Predicting volume distributions of hardwood sawn products by tree grade in eastern Canada

Predicting volume distributions of hardwood sawn products by tree grade in eastern Canada Abstract Northern hardwoods are an ecologically and economically important forest type in eastern North America. Historically, the hardwood supply came from old-growth forests dominated by large-diameter trees. Unfortunately, the repeated removal of high-quality trees has substantially degraded hardwood forests and reduced the profitability of the primary manufacturing sector. In this context, forest managers need tools to guide silvicultural investment decisions and to estimate pre-harvest stand value based on forest inventories. The objective of this study was to evaluate the performance of classification systems and measured variables used at the tree level to predict sawn product volumes of sugar maple (Acer saccharum Marsh.), yellow birch (Betula alleghaniensis Britton) and American beech (Fagus grandifolia Ehrh.). We developed statistical models to estimate the volume of lumber products, pulpwood, sawdust and residues based on tree DBH, species, tree grades in different combinations and tree height. Results show that the tree grade variable increased the explained variation in product volumes. As expected, the accuracy of product volumes estimation, based on root mean square error (RMSE), was poor for an individual tree, but improved as the number of trees increased. Introduction The northern hardwood forest is a major forest type in eastern Canada. In Québec, it represents 16.3 per cent of the total productive forest landbase and 22.5 per cent of the total gross merchantable volume (MFFP, 2015). The province’s hardwood sawmills generate direct economic outcomes similar to the softwood industry, though at a smaller production scale (Trudelle et al., 2009). High-grade lumber is mainly used for appearance-based products such as furniture, flooring, cabinets and moudlings, while lower grades are used in packaging products (e.g. pallets). Lower-grade hardwoods can also be used for new structural applications (e.g. in cross-laminated timber for non-residential multistory buildings; Gong et al., 2015). Historically in eastern Canada, the hardwood supply came from old-growth forests where large-diameter trees were abundant. Despite limited knowledge about the resource and with only simple processing technologies, it was relatively easy to extract high-value products from those trees. However, over the years, the repeated removal of high-quality trees has substantially degraded hardwood forests and reduced the industry’s profitability to marginal levels. In this context, provincial forest managers need detailed knowledge of the current quality and value of hardwood forests to justify silvicultural investments and wood allocation. They need tools or models that can estimate the quantity and quality of products at the tree or stand level (i.e. enhanced forest inventory and tree marking processes that lead to a better estimate of pre-harvest stand value). In the past few decades, several tree classification systems have been developed in the United States (USA; Hanks, 1976a), and eastern Canada (Majcen et al., 1990; MRN, 1995; OMNR, 2004; Pelletier et al., 2013). These systems can be divided into two main groups: (1) tree grading (TG) classification systems, which aim to evaluate the tree’s potential to produce lumber products (e.g. Hanks, 1976a; MRN, 1995) and (2) hybrid Risk-Product (RP) classification systems, which comprise components both for tree mortality risk and for the stem’s potential product (e.g. Majcen et al., 1990; OMNR, 2004). Tree grading systems are generally used in national forest inventory programs to evaluate standing wood quality (USDAFS, 2012; MFFP, 2016). Hybrid tree classification systems were mainly developed for pre-harvest inventories and for tree marking. They aim to evaluate a tree’s potential for lumber products and its harvest priority. Tree grading evaluation is generally too time-consuming to be a standard part of pre-harvest inventories. In eastern Canada, some models were developed to better estimate the sawlog potential of standing trees (Fortin et al., 2009; Havreljuk et al., 2015). These models estimate the net volume that a tree can generate in each log grade, but do not allow a direct volume estimation of lumber products by visual grade. Other research has also examined how net product value per tree varies within a range of defects or tree grades but they did not develop models to estimate the volume of each lumber product in a tree (Cockwell and Caspersen, 2014; Havreljuk et al., 2014; Cecil-Cockwell and Caspersen, 2015). To determine the potential value of a standing tree, we need to estimate a tree’s potential to produce lumber in current markets, which generally consider lumber by quality grades (NHLA, 2007) using a price list (e.g. Hardwood Market Report: www.hmr.com). To our knowledge, only Hanks (1976b) has developed lumber models at the tree level based on tree grades in the USA. However, we hypothesized that these may not be applicable in stands located near the northern edge of their climatic range in eastern Canada. In this region, trees are exposed to extreme climatic conditions that promote stem defects such as frost cracks, which in turn reduce lumber value (Kubler, 1983; Burton et al., 2008). This study evaluated the performance of two tree classification systems to predict the volume of lumber grades, pulpwood, sawdust and residues for the three most important tree species found in northern hardwood forests: sugar maple (Acer saccharum Marsh.), yellow birch (Betula alleghaniensis Britton) and American beech (Fagus grandifolia Ehrh.). By predicting product volumes, our proposed approach allows users to adjust the monetary value of any product for a given market and obtain an assessment of current stand value. We also tested whether tree classes or tree grades could be merged to simplify the systems and make the estimations less time-consuming. Methods Study site The study was carried out in the Duchesnay Forest (lat. 46° 57′ N, long. 71° 40′ W) near Québec City, Canada. This forest is located in the meridional subregion of the balsam fir (Abies balsamea [L.] Mill.)–yellow birch bioclimatic domain (Robitaille and Saucier, 1998). Depending on elevation and aspect, the vegetation is characterized either by yellow birch–balsam fir stands or by sugar maple–yellow birch stands. Stands selected for this study are dominated by yellow birch, sugar maple and American beech (which, respectively, represent 38 per cent, 34 per cent and 23 per cent of merchantable basal area, i.e. that of trees with a diameter at breast height [DBH] larger than 9 cm). Podzolic soils in the region developed from deep glacial till. The bedrock is mainly composed of granitic gneiss from the Grenville Province. According to the climate estimations generated by BioSIM for the 1980–2009 period (Régnière et al., 2012), the mean annual temperature, and the mean annual maximum and minimum temperatures at the study site, are 3.0°C, 30.4°C and −35.3°C, respectively. Sampling and tree assessment A total of 52 yellow birch, 52 sugar maple and 35 American beech trees were sampled mostly from those marked for cutting in eight experimental units (1.2 ha) distributed within two different blocks of a study set up to compare different rehabilitation silvicultural treatments (Bédard et al., 2014). In some of these units, especially for the highest grades, it was necessary to select unmarked trees in the buffer zone surrounding the experimental units (20 m around the central plot) in order to cover both the desired range of DBH (i.e. 24–70 cm) and tree grades. Despite this, certain tree grades were very scarce and yielded smaller samples than others (e.g. grade A, Table 1). Total tree height and merchantable tree height (taken at the base of the crown’s main branches) were also estimated using a Vertex III (Haglöf, Sweden AB). Mean total tree height (±1 standard deviation) was 22.5 ± 3.2 m for yellow birch, 21.1 ± 3.0 m for sugar maple and 21.7 ± 2.8 m for American beech; mean merchantable tree height for these species was 13.0 ± 2.5 m, 13.2 ± 2.0 m and 13.0 ± 2.1 m, respectively. Before felling, trees were numbered, and species and DBH were recorded. Each tree was also classified according to the two classification systems described below. Table 1 Number of trees harvested, mean, standard deviation and range of tree DBH, by species and tree classification system. Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Note: n = number of trees harvested for each tree classification group for every species; SD = standard deviation; TG = tree grade classification, grade A being the highest quality and grade D, the lowest; RP = risk-product classification; AGS = low mortality risk, sawlog potential; UGS = high-mortality risk, sawlog potential; Cull = low or high-mortality risk, pulpwood or firewood potential. Table 1 Number of trees harvested, mean, standard deviation and range of tree DBH, by species and tree classification system. Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Note: n = number of trees harvested for each tree classification group for every species; SD = standard deviation; TG = tree grade classification, grade A being the highest quality and grade D, the lowest; RP = risk-product classification; AGS = low mortality risk, sawlog potential; UGS = high-mortality risk, sawlog potential; Cull = low or high-mortality risk, pulpwood or firewood potential. The TG classification system separates trees into four grades (A, B, C and D) according to their DBH and their external defects on the best 3.7-m stem section within the lower 5 m (Figure 1). Grade A stems represent the best quality, i.e. trees with the highest sawlog potential in the lower 5-m section, while grade D stems have no sawlog potential. Trees must meet a specific minimal DBH for each grade (A: > 39 cm, B: > 33 cm, C and D: > 23 cm). External defects are assessed in order to estimate the length of the defect-free sections on each of the four faces. The yield of the third-best face and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade (Figure 1). The defect-free section must total at least 3.1, 2.5 and 1.8 m in length for grades A, B and C, respectively. It could be in one or two sections of at least 1.5 m in length for grade A, in one to three sections of at least 1 m in length for grade B, and in one to three sections of at least 0.6 m in length for grade C. Grades A, B and C denote trees with potential sawlogs and are similar to tree grades 1, 2 and 3 used by Hanks (1976a) in the USA. Grade D is equivalent to the ‘below grade’ in Hanks’ classification, and indicates possible local use of bolts or pulpwood. The main differences between tree grades used in Québec and in the USA are the conversion from imperial to metric units and the absence of a minimum diameter inside bark at the top of the grading section. Figure 1 View largeDownload slide Standing tree classification system based on the best 3.7-m stem section within the lower 5 m. External defects are assessed in order to estimate the length of the defect-free sections on each four faces (tree circumference is divided into four equal parts). The yield of the third-best face, tree DBH and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade. Figure 1 View largeDownload slide Standing tree classification system based on the best 3.7-m stem section within the lower 5 m. External defects are assessed in order to estimate the length of the defect-free sections on each four faces (tree circumference is divided into four equal parts). The yield of the third-best face, tree DBH and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade. The hybrid RP classification system encompasses two components: tree mortality risk and a stem’s potential product. Each component is assessed using a two-level classification developed for tree marking in selection cuttings (Majcen et al., 1990). The RP classification aims to identify trees that are at high risk of mortality or of losing stem quality before the next cutting cycle (i.e. 15–25 years). Basically, it is based on crown, root and bole defects. If more than one-third of the crown shows evidence of dieback or damage, or if some major defect can be observed (e.g. presence of fungi, canker, decay, major root damage or large open wounds), the tree is assigned to the high-mortality risk class. Otherwise, it is considered as having a low risk of mortality (see Guillemette et al., 2008 for more details). Likewise, depending on bole straightness and external defects, hardwood trees are classified as pulpwood or potential sawlogs. The minimum requirement for potential sawlogs was the presence of a 1.8-m log, at any location on the stem, with one face free of defects and with no deduction for rot applicable to this grading section. The RP classification comprises four classes: (1) low mortality risk–sawlog potential, (2) low mortality risk–pulpwood or firewood, (3) high-mortality risk–sawlog potential and (4) high-mortality risk–pulpwood or firewood. However, these classes were grouped into three categories: (i) acceptable growing stock (AGS), which corresponds to the first class, (ii) unacceptable growing stock (UGS), equivalent to the primary class 3 and (iii) cull, composed of classes 2 and 4. These categories are often used in pre-harvest forest inventories, in marking guides and for silvicultural prescriptions in hardwood forests of eastern North America (OMNR, 2004; Leak et al., 2014; OMNRF, 2015). Bucking and sawmilling The selected trees were felled and topped by a harvester according to a whole-tree harvesting system and brought to the forestry school lumber mill in Duchesnay, Québec. After visual assessment of external characteristics, each stem was bucked into logs following the procedure described by Petro (1971) in order to maximize the production of high-grade lumber products (NHLA, 2007). Each log was then graded (F1, F2, F3, from the highest to the lowest quality) for factory lumber (Rast et al., 1973; Petro and Calvert, 1976), local-use bolt (F4) or pulpwood (P). The main grading factors for sawlogs are their position in the stem, length, small-end diameter, end defects, length and number of defect-free sections (clear cuttings), and scaling deductions for sweep, crook or rot. While the smallest sawlog must be at least 2.5 m long with a diameter of 20 cm at the small end, the local-use bolt grade accepts lengths down to 1.9 m and diameters down to 16 cm. The minimum requirement for pulpwood was a 2.44-m long, relatively straight piece with a small-end diameter (SED) of at least 9.1 cm. Some cull pieces, for which conversion to products was impractical (Rast et al., 1973), were classified as residues after bucking. The 139 sample trees were cut into 408 logs, of which 139 were classified as sawlogs. The 139 sawlogs were sawn into 1762 boards (yellow birch: 818; sugar maple: 656; American beech: 288) and 128 central blocks (yellow birch: 54; sugar maple: 52; American beech: 22), using a sawing-around cutting pattern. This process consists in feeding a carriage with a log, aligning the log against a saw blade to open the best possible face, extracting one or more boards, and then rotating the log at 90° before extracting another board. This operation continues until a central block (pallet cant) of lower visual quality is obtained. The number of rotations around the log depends on its size, form, internal colouration and defects. This selective sawing aims to maximize the production of defect-free boards in the high value, light-coloured sapwood part of the log. Sawn boards were 1.22–3.66 m (4–12 ft) long, 7.6–33.0 cm (3–13 inches) wide and 4/4 (25.4 mm or 1 inch) thick. In reality, the sawmill processed each board with an overthickness of 2.54 mm (0.1 inch). This common practice allows for product finishing/surfacing (e.g. planing, sanding). Thus, the actual board thickness used to calculate lumber volumes in this study was 27.94 mm (1.1 inches). Boards were graded in the rough green state under the National Hardwood Lumber Association’s grading rules (NHLA, 2007), which are based on the percentage of clear defect-free wood on a board. Central blocks (dimension: 4 × 4 inches, sawn without overthickness) were not further transformed into pallet wood at the sawmill. A complete tree–log–lumber tracking was kept throughout the study. The volume of all logs was calculated with Smalian’s formula, using the diameter and length data recorded at bucking. Lumber products were compiled in number, volume and quality for each log and each tree. Lumber volume (length × width × thickness) was calculated in imperial units and then converted into metric units (m3) to be consistent with the other tree product volumes. Sawdust volume was estimated as 15 per cent of the sawlog volume (average value of all sawlogs) based on Optitek sawing simulations (FPInnovations, 2014). Overall pulpwood volume per tree was calculated by adding the volume of logs classified as pulpwood at bucking (at least 2.44 m long and 9.1 cm in SED) to the volume of pulpwood produced at the sawmill (i.e. slabs and poor quality lumber below NHLA grades). The volume of ‘sawmill-generated’ pulpwood per log was estimated by subtracting lumber and sawdust volumes from the gross sawlog volume. When a sawlog contained decay, the estimated volume of decayed wood was added to the sawdust volume previously estimated using Optitek, since pulp mills usually do not want decayed pulpwood (chips) in their supply. Model development The models’ dependent variables correspond to the net volume (m3) of each product within a tree. Nine different product types were retained, the first five being groups of NHLA grades (NHLA, 2007) defined as Select (select sap, select red, select), No. 1 Common (1Com), No. 2 Common (2Com), No. 3 A Common (3ACom) and No. 3B Common (pallet, 3BCom); these were followed by blocks, pulpwood, residues and sawdust. Table 2 presents descriptive statistics for net volumes by species and product type and Figure S1 in the Supplementary Material shows the observed net volumes for lumber and non-lumber products as a function of DBH for each species. Table 2 Number of trees that produced a final product, mean, standard deviation and range of tree volume, by species and product type. Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Note: n = number of trees of each species that produced volume of each product; SD = standard deviation; Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3 A Common; 3BCom = No. 3B Common (pallet). All trees produced pulpwood while some trees did not produce any lumber products. Table 2 Number of trees that produced a final product, mean, standard deviation and range of tree volume, by species and product type. Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Note: n = number of trees of each species that produced volume of each product; SD = standard deviation; Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3 A Common; 3BCom = No. 3B Common (pallet). All trees produced pulpwood while some trees did not produce any lumber products. Since volume distributions showed an excess of zero values for most products, we used a two-part conditional model (Cunningham and Lindenmayer, 2005). The first part of the model predicted whether product j was observed in tree i. If this was the case, the volume of product j was then predicted by the second part of the model, which excluded all zero values. The final volume predictions, or marginal volumes, were then obtained by multiplying the probability of occurrence predicted by the first part and the conditional volume predicted by the second part. For the first part, the probability of observing product j in tree i ( pij) was obtained using a generalized linear model with a logit link function: log(pij1−pij)=Zijγ, (1) where Zij is a row vector of explanatory variables related to product j of tree i, and γ corresponds to the vector of unknown fixed-effects parameters. The second part of the model predicted volume vij, i.e. the volume of product j in tree i, given that vij > 0, so conditional on the presence of product j in tree i (i.e. vij|pij). The volumes were square root transformed to achieve normality and to prevent the prediction of negative volumes. The vector of within-tree square root transformed volumes ( yi=vi, with vi being the vector of vij) was assumed to follow a multivariate normal (MVN) distribution: yi~MVN(Xiβ,Ri), (2) where Xi is a matrix of independent variables for tree i, β is a vector of unknown fixed-effects parameters, and Ri is the generalized variance–covariance matrix of the within-tree error terms (UN, unstructured), used to take into account possible correlations between the products within a tree. The predicted square root transformed volumes were back-transformed to their original scale after correcting for the bias induced by the transformation (Gregoire et al., 2008): E(vi|pi)=(Xiβˆ)2+diag(Rˆi), (3) where E(vi|pi) is the mathematical expectation of vector vi, and diag(Rˆi) corresponds to the diagonal elements of Rˆi, which are the variance estimates of each product ( σˆj2). Marginal predictions of the volumes were finally obtained by multiplying the conditional predictions by the probabilities of observing the products (equation [1]): E(vij)=E(vij|pij)×pij. (4) Model specification and diagnostic The explanatory variables used to formulate the candidate models were species, DBH, total tree height (TH), merchantable tree height, TG classification (A, B, C, D) with different grade groupings, and RP classification (AGS, UGS, Cull). Interactions between these variables were also tested. For the first part of the model predicting the presence of the products, the parameters of significant variables were estimated using the maximum likelihood (ML) method in the SAS LOGISTIC procedure (SAS Institute Inc., 2013) with the logit link function. Firth’s bias correction (Firth, 1993) was used to reduce bias in the parameter estimates and to solve convergence problems. For the second part of the model, significant variables and the estimates of the final models were determined using the REML (restricted maximum likelihood) method in the SAS MIXED procedure. The various candidate models for both parts (probability of occurrence and conditional volume) were then compared and ranked using the Akaike (AIC and AICC) information criteria (Burnham and Anderson, 2002) obtained with the ML (maximum likelihood) method: the smaller the value of AIC, the better the model. The area under the receiver operating characteristic curve (AUC) (Allison, 2012) was also used to determine the best candidate model for the probability of occurrence. We considered that an AUC > 0.80 indicated that the model was accurate. The Hosmer–Lemeshow test (Hosmer and Lemeshow, 2000), for which a significant probability related to the χ2 statistic indicates a lack of fit, was also used. Normality and homogeneity of variance of the conditional volume model were checked visually with normalized (scaled) plots of residuals taking into account the autocorrelation between products, expressed by the R generalized variance–covariance matrix (equation [2]). Model evaluation The performance of each model’s predictions was evaluated at the tree level by a leave-one-out cross-validation. Every model was fitted 139 times, omitting one tree and modelling the probability of occurrence or the conditional volumes of the 138 other trees. The probabilities of occurrence and the conditional volumes of every product for the left-out tree were predicted with the resulting models. We performed a Hosmer–Lemeshow test on the predictions of the first part of the model, whereas we computed biases and root mean square errors (RMSE, as presented in Weiskittel et al., 2011) for every product and species based on predicted conditional volumes resulting from the cross-validation for the second part, after having discarded the null volumes. Moreover, we calculated biases and RMSE on marginal volume predictions obtained by multiplying the conditional volume predictions of each product of every tree by the corresponding predicted probabilities of observing the products. Finally, we calculated relative biases and relative RMSE by dividing the resulting values by the mean observed volume. We also assessed model performance for a larger number of trees based on subsets of randomly selected trees, in order to verify whether accuracy (RMSE) was improved. This was calculated from a leave-k-out cross-validation in which 5–45 trees (by increments of 5) of the 139 trees were removed from the calibration dataset at each iteration. We stopped at a maximum of 45 trees, which represented one-third of the sample trees. The remaining two-thirds or more of the sample were used to calibrate both parts of the model. Marginal volume of every product of each left-out tree was then calculated by multiplying the probability of occurrence with the corresponding conditional volume. The mean of the observed and predicted marginal volumes of every product for each species of the left-out group were calculated. We repeated this procedure 1000 times, and calculated biases and RMSE on the 1000 resulting mean values. Results Several candidate models were tested for both parts of the statistical model; the best candidate models for each combination are presented in Tables 3 and 4. Although other models that included significant interactions between tree grade and species or DBH showed smaller log-likelihood values than the best candidate models, they were not retained because the higher value of their information criteria indicated that simpler models were better. We also tested total tree height and merchantable height, alone and in interaction with other variables, but the resulting models were not better than those presented. Table 3 Model comparison of the probability of occurrence of a particular product (first part of the model) based on Akaike criteria value (AIC), area under the curve (AUC) and Hosmer and Lemeshow’s goodness-of-fit test. Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Note: Rank is the model ranking according to Akaike criteria value (AIC). Table 3 Model comparison of the probability of occurrence of a particular product (first part of the model) based on Akaike criteria value (AIC), area under the curve (AUC) and Hosmer and Lemeshow’s goodness-of-fit test. Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Note: Rank is the model ranking according to Akaike criteria value (AIC). Table 4 Model comparison of the conditional volume of a particular product (second part of the model) based on Akaike criteria values (AIC and AICC). Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Note: Rank is the model ranking according to Akaike criteria values (AIC and AICC). Table 4 Model comparison of the conditional volume of a particular product (second part of the model) based on Akaike criteria values (AIC and AICC). Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Note: Rank is the model ranking according to Akaike criteria values (AIC and AICC). For the first part, regarding the probability of occurrence of each product, the simplest model (Model 1) had the largest AIC value as well as the lowest AUC value, indicating that all other models were better to predict the occurrence of a particular product in a given tree. AUC values were >0.80 for all models, indicating that they all were accurate. In addition, all models had nonsignificant Hosmer–Lemeshow statistics at the 1 per cent significance level. Based on AIC values, the best models were those that included DBH and tree grade (TG, Models 2, 3 and 6). The best one was Model 6, which included DBH, species and TH. Its AIC value was 320 units lower than that of Model 1. Equation [1] is then: log(pij1−pij)=Zijγ=γ1,jDBHi+γ2,jDBHi2+γ3,jq+γ4,jTHi, (5) where pij is the probability of observing product j in tree i, Zij is a row vector of explanatory variables associated with product j in tree i, q corresponds to the tree grade index ( q=1,2,3 for grades AB [A or B], C and D, respectively). The parameter estimates of this equation and their standard errors are presented in Table 5. For example, the probability of observing the Select product for a grade A tree with a DBH of 50 cm and a total tree height of 24 m is: pSelect=e(−9.6641+0.2773*50−0.00325*502+0.2561*24)(1+e(−9.6641+0.2773*50−0.00325*502+0.2561*24))=0.9022. (6) Table 5 Parameter estimates and standard errors (in parentheses) of the best model predicting occurrence of each product, by product type (Model 6: DBH, DBH2, tree grade [AB, C, D] and total tree height [TH]). Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). γˆ1,j: Parameter estimate of each product for DBH effect. γˆ2,j: Parameter estimate of each product for DBH2 effect. γˆ3,j1 , γˆ3,j2 , γˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. γˆ4,j: Parameter estimate of each product for TH effect. Table 5 Parameter estimates and standard errors (in parentheses) of the best model predicting occurrence of each product, by product type (Model 6: DBH, DBH2, tree grade [AB, C, D] and total tree height [TH]). Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). γˆ1,j: Parameter estimate of each product for DBH effect. γˆ2,j: Parameter estimate of each product for DBH2 effect. γˆ3,j1 , γˆ3,j2 , γˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. γˆ4,j: Parameter estimate of each product for TH effect. From the parameter estimates presented in Table 5, predicted probabilities of occurrence were generated. They are illustrated in Figure S2 (Supplementary Material) for two different tree heights corresponding to the first and third quantiles of tree height distribution of all sampled trees. Overall, the probability of obtaining lumber products increased with TG and TH. It also increased with DBH until 40 cm, and then decreased with larger DBH values. For non-lumber products, the probability of occurrence for pulpwood was near 100 per cent and did not vary with TG, TH or DBH. For residues, the probability increased with DBH and TH and was similar between tree grades AB and C, but lower for tree grade D. For sawdust, the probability increased with both TH and TG. It also increased with DBH until 40 cm, then decreased for larger DBH values. The Hosmer–Lemeshow’s goodness-of-fit test performed on predictions resulting from cross-validation was significant ( χ2 = 29.16 and P = 0.0003), but a visual comparison of observed and predicted probabilities showed no major problem, neither over all species nor by species (not shown). For the prediction of conditional volumes, the simplest model (Model 1) had the largest AIC and AICC values (Table 4). Therefore, all other models performed better than Model 1 to predict the conditional volume of a particular product in a given tree. Based on the Akaike criteria values, the best models were those that included DBH, species, and TG (Models 2 and 3). Model 3, which included DBH, species, and TG with grades AB, C and D, ranked best (with AIC and AICC values that were, respectively, 62 and 56 units lower than those of Model 1). Model 2, equivalent to Model 3 but without a grouping of grades A and B, ranked second. Considering Model 3 as the best one, the right-hand side of the model of equation [2] can be expressed as: Xijβ=β0,js+β1,jDBHi+β2,jDBHi2+β3,jq, (7) where Xij is a row vector of explanatory variables associated with product j in tree i, s is the species index ( s=1,2,3, with the three levels referring to yellow birch, sugar maple and American beech, respectively), and q corresponds to the tree grade index ( q=1,2,3 for the AB [A or B], C and D tree grades, respectively). The parameter estimates of this equation and their standard errors are presented in Table 6, along with the variance estimates of each product that are needed to transform volumes back to their original scale. For example, the conditional volume of Select lumber for a grade A yellow birch tree with a DBH of 50 cm is: vSelect=(−0.5349+0.02965*50−0.00023*502+0.06623)2+0.00945=0.2020m3. (8) Table 6 Parameter and variance estimates and standard errors (in parentheses) of the best conditional volume model, by product type (Model 3: species, DBH, DBH2 and tree grade [AB, C, D]). Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). Diag(Rˆi) is the diagonal of the generalized variance–covariance matrix. βˆ0,j1, βˆ0,j2, βˆ0,j3: Parameter estimates of each product for yellow birch, sugar maple and American beech, respectively. βˆ1,j: Parameter estimate of each product for DBH effect. βˆ2,j: Parameter estimate of each product for DBH2 effect. βˆ3,j1, βˆ3,j2, βˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. σˆj2: Variance estimate of each product. Table 6 Parameter and variance estimates and standard errors (in parentheses) of the best conditional volume model, by product type (Model 3: species, DBH, DBH2 and tree grade [AB, C, D]). Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). Diag(Rˆi) is the diagonal of the generalized variance–covariance matrix. βˆ0,j1, βˆ0,j2, βˆ0,j3: Parameter estimates of each product for yellow birch, sugar maple and American beech, respectively. βˆ1,j: Parameter estimate of each product for DBH effect. βˆ2,j: Parameter estimate of each product for DBH2 effect. βˆ3,j1, βˆ3,j2, βˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. σˆj2: Variance estimate of each product. From the parameter estimates presented in Table 6, we generated conditional volume predictions and then corrected them for the bias induced by transformation using the correction in equation [3]. The resulting predicted values are illustrated for each species in Figures S3–S5, for two different tree heights corresponding to the first and third quantiles of tree height distribution of each species. Overall, the predicted conditional volumes of most of the lumber grades increased with DBH and TG for each species. However, the conditional volume of blocks increased with DBH up to 40–45 cm and then decreased for larger DBH values for each TG and species. Conditional volume of the Select lumber grade was larger for yellow birch, followed by American beech and sugar maple. For other lumber grades, the predicted conditional volumes were similar. For non-lumber products, the conditional volume increased with DBH and decreased with tree quality. However, the conditional volume of pulpwood was higher for American beech than for the two other species. The conditional volume of residues and sawdust were similar between species. Regarding the leave-one-out cross-validation for the prediction of conditional volumes, relative bias values were variable, ranging from −30.0 per cent to 25.2 per cent, but mostly between −8.8 per cent and 8.0 per cent (Table S1). Note that negative bias indicates an overestimation, whereas a positive bias represents an underestimation. Regarding accuracy, relative RMSE values varied from 36.0 per cent to 124.9 per cent (Table S1). Finally, from the retained two-part model, marginal volume predictions were generated by multiplying the conditional volume predictions by the probabilities of observing the products. The resulting predicted values are illustrated in Figures 2–4, for two different tree heights corresponding to the first and third quantiles of tree height distribution of each species. Marginal lumber volume generally increased with DBH, TG and TH (Figures 2a, 3a and 4a). As expected, tree grade AB produced larger volumes of high-quality lumber products for a given height (Select, 1Com) as well as larger volumes of medium-quality (2Com), 3ACom and blocks than tree grades C and D. Among the five lumber grades, volumes of 3BCom were the lowest produced for all species, and this volume was similar between tree grades. Volume differences between the highest and the lowest lumber grades decreased along with tree grades: the largest differences between lumber grades were observed for grade AB trees and the smallest differences, for grade D trees. Predicted volume of the Select grade was greater for yellow birch than for other species. Pulpwood volume also increased with DBH, but contrary to lumber volume, it decreased as either tree grade or tree height increased (Figures 2b, 3b and 4b). Pulpwood volume was similar for yellow birch and sugar maple, but slightly greater for American beech for all tree grades. Sawdust and residue volumes increased with DBH, and also increased very slightly with tree grades. For each tree grade, sawdust and residue volumes were similar between species. Regarding the leave-one-out cross-validation, the relative biases for products were variable, ranging from −58.8 per cent to 20.7 per cent (Table 7). The relative biases were small for pulpwood for all species, as well as for yellow birch 3ACom and 2Com products, for sugar maple Select, 3ACom and residues, and for American beech residues (−6.3 per cent to 6.6 per cent). However, relative biases were rather large for other lumber grades for sugar maple and yellow birch (−40.9 per cent to 20.7 per cent). The biases obtained for American beech were generally larger than for other species. The biases obtained with 45-tree subsets were generally similar to those obtained at the tree level (Table 7). Regarding accuracy, the relative RMSE decreased as the number of trees in the subset increased, which means that volume estimates were much more accurate for larger subsets of trees (Figure S6 in Supplementary Material). For example, the relative RMSE values were much lower for the 45-tree subsets (15.3–101.5 per cent, Table 7) than at the tree level (44.9–292.9 per cent). Table 7 Results of the cross-validation (bias and root mean square errors [RMSE], both in absolute and relative values) for the best marginal volume model (i.e. Models 6 and 3) Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Note: n = Number of observations used for calculations. Mean volumes were calculated using all observed volumes, including null volumes for each product type. Relative RMSE values for different tree subsets are also presented in Figure S6. Table 7 Results of the cross-validation (bias and root mean square errors [RMSE], both in absolute and relative values) for the best marginal volume model (i.e. Models 6 and 3) Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Note: n = Number of observations used for calculations. Mean volumes were calculated using all observed volumes, including null volumes for each product type. Relative RMSE values for different tree subsets are also presented in Figure S6. Figure 2 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing yellow birch trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 2 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing yellow birch trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 3 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing sugar maple trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 3 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing sugar maple trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 4 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing American beech trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 4 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing American beech trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Discussion From a statistical viewpoint, both tree classification systems analysed in this study improved the prediction of both the occurrence and the conditional volume of a given product compared with the simplest model (Model 1, Tables 3 and 4). Adding RP classes (Model 4, Tables 3 and 4) slightly improved occurrence and conditional volume estimates, but TG models, with TH for the first part, ranked best (Model 6 in Table 3 and Model 3 in Table 4). For better accuracy, the volume of each product type should therefore be estimated using both the TG classification and a measure of TH. However, from a practical viewpoint, it is not reasonable to measure TH on each tree of the sample plot during a pre-harvest forest inventory. It is more realistic to estimate TH using a local height-diameter relationship. Merging tree grades A and B in TG models resulted in better models (Models 3 and 6 in Table 3 and Model 3 in Table 4), but this could be due to the small sample size for grade A trees (Table 1). In any case, this situation reflects the reality of the high-graded northern hardwood forests, in which grade A trees are sparse. We also evaluated further simplifications of the two classifications by merging of grades. None were statistically better (based on AIC values) than Models 6 and 3, which contained grades AB (i.e. A and B grouped), and grades C and D (modelled separately), TH in the first part and species in the second part of the model. From a practical viewpoint, the most important advantage of using the TG classification is its capacity to discriminate stems with a better potential to produce higher-quality sawn products, compared with models which do not include tree quality assessment. This result indicates that better tree grades are not solely related to larger DBH, but also to external tree characteristics that influence internal wood product quality. As we mentioned before, biases were large for some products (Table 7). The best two-part conditional model (Model 6 for the first part of the model and Model 3 for the second part) was less accurate for predicting volumes for 3BCom, probably because of the smaller sampled volume for this grade. For American beech, biases were also large for most other grades, probably because of the smaller number of trees sampled compared to the other species. The cross-validation showed that using 45-tree subsets generally improved results for all product categories, even if the biases and RMSE values remained high for some products (Table 7). We tried to reduce bias and improve accuracy by grouping product categories (e.g. high [Select, 1Com, 2Com] and low lumber grades [3ACom, 3BCom, Blocks]), but we obtained similar results. Grouping product categories has the additional disadvantage of reducing the accuracy of tree value assessments, given the significant price differences for some products, especially in higher grades. Assigning an average price to higher grades (e.g. Select, 1Com and 2Com) appears questionable if, for example, the proportion of these grades are not known, because it could result in an under- or overestimation of total tree value. To reduce the biases of the best model, we would have needed to sample more trees for each species and tree grade. As expected, the accuracy of the estimated volume was low at the tree level. Although trees can be grouped according to species, DBH and tree grade following a forest inventory to evaluate potential products, a large portion of among-tree variability in product yields remained unexplained, resulting in a very low accuracy at the tree level (Table 7). Nonetheless, when the objective is to estimate the value of standing hardwood timber prior to cutting based on a pre-harvest forest inventory, high accuracy is not required at the individual tree level. Our simulations showed that the relative accuracy (RMSE) improved with the number of trees retained in the subsets (Figure S6, Supplementary Material). If we assume that relative accuracy would continue to improve (to some extent) for subsets of more than 45 trees, we can expect more accurate yield predictions for a larger number of trees. This hypothesis could not be tested due to the limited sample size. Moreover, we did not evaluate the accuracy at the stand level. Biases for any particular stand type or site may also exist. For example, Germain et al., 2015 found that dark heartwood size of sugar maple trees varied between sites in New York State. They found that sites with less acid soils had fewer large dark hearts, and consequently trees of higher value because high value of sugar maple lumber depends on the wood being light-coloured. The marginal lumber volumes predicted for yellow birch and sugar maple with our best two-part conditional model were much smaller than those predicted by Hanks (1976b) in the USA. The difference generally decreased with DBH for each grade. For example, for a grade 3 yellow birch stem (similar to tree grade C in our model) with a merchantable height of 13 m (43 feet), a total height of 22.5 m, our model predicted a volume for 1Com that represents 17 per cent of the volume predicted by Hanks for a DBH of 30 cm, and only 25 per cent for a DBH of 40 cm. The frequent observations of defects on the bole of our study trees, such as cracks, seams and dead branches (data not shown), combined with the common presence of a dark heartwood in sugar maples, could explain the smaller volumes of lumber grades. Compared with the study area of Hanks (1976b), ours is located at the northern limit of the distribution range of northern hardwoods, where lower stem quality can be expected. Indeed, we observed an apparently lower potential to produce quality lumber and a higher potential for pulpwood products in our study area. In a similar study in the same bioclimatic domain in which paper birch trees (Betula papyrifera Marsh.) were graded by harvesting priority based on mortality risk, Drouin et al. (2010) showed that DBH was the most important variable affecting lumber quality and value, followed by tree harvest priority. Larger trees were associated with higher lumber quality and higher lumber value in $/m3 per tree. This trend supports silvicultural treatments aimed at producing larger trees. However, the proportion of discoloured wood increases as trees grow older (Baral et al., 2013; Duchesne et al., 2016) potentially causing value loss. A more balanced approach may be needed, depending on the ability of each species to recover from traumatic events (wounds, fungal attack, etc.) that cause discoloration (i.e. to find a trade-off between tree size, discoloration and mortality risks for each species). Another study on paper birch established that larger and less vigorous trees produced boards with higher proportions of discoloured wood (Drouin et al., 2009); the effect of tree age was indirect through tree diameter. These results illustrate the complex interplay or dynamics between tree diameter growth, vigour and mortality risk caused by biological processes affecting colour and soundness of wood, which, altogether, influence timber value. It also highlights the need to have better operational tools to predict when trees have reached their maximum value (Pothier et al., 2013; Guillemette, 2016). In a value chain perspective, the poor quality of hardwood stands could make tree grading and marking economically unviable. If we assume that tree quality is sufficiently good to cover the costs of tree grading, then we think that tree grading should primarily focus on identifying only high- and low-potential trees, in order to lower costs. High-potential trees would be those with a DBH > 23 cm having at least the defect-free sections and volume reductions of grade B, since grade AB yielded much larger marginal volumes of higher grade products than other tree grades. Low-potential trees would be trees that do not qualify for grade B. This dichotomic approach would simplify tree grading and reduce the time required to grade, because only the criteria for grade B would be applied in the field. In order to evaluate actual stand value in northern conditions where grade A trees are very scarce and often hide overgrown internal defects, we could rely on evaluating only high and low grades on larger trees (i.e. DBH > 33 cm). However, in order to evaluate future stand quality, one must also consider high-potential trees in the smaller diameter classes (i.e. DBH 23–33 cm) especially in uneven-aged stands. Such crucial information on the potential quality of the forests to come (i.e. regarding the quality of tree recruitment, by grading high-potential trees) would strongly support sustainable forest management. Knowing the forest value could be more critical for decision-making in hardwood stands located at the northern limit of the species’ distribution range than on more southern sites. Indeed, in the sites studied by Hanks (1976b) in the USA, trees were more likely to produce lumber and generate revenues. The incentive for grading trees will also depend on market price differences between lumber grades, and on the demand for transforming the large volumes of pulpwood (which can represent, on average, 65 per cent of the volume of the trees analysed) associated with lumber production. The higher the premium for high-quality grades (Select, 1Com), the stronger the motivation could be to mark trees that are likely to produce these grades. Ultimately, stand value assessment and harvest decisions will depend on silvicultural and economic objectives. Though the subject needs more investigation, this study provides some useful insights. Conclusion The models developed in this study should be considered as a step toward better understanding the relationships between hardwood tree classification systems and their volume yield in sawn products. By modelling volume distributions of sawn products instead of total tree value, lumber product and pulpwood prices can be adjusted to any given market of interest to reflect current stand value. Our results indicate that sawn product volumes are better estimated after an initial assessment of stem quality. A system such as the tree grade (TG) system, possibly simplified by using only two categories (high- and low-potential trees), could help produce high-quality lumber grades and reduce the cost of forest inventories. Thus, product models such as those presented in this study could assist in decision-making, by simulating the impact of different forest management scenarios on stand value (e.g. when related to growth and yield models) and by estimating sawn product volume distributions and value in auctioned stands before harvest. However, additional sampling would be needed to test the hypothesis that the accuracy of our models improves at the stand scale and to allow the testing of site differences. Given their influence on product yields, various tree bucking procedures and log cutting patterns used in sawmills should also be compared. Supplementary data Supplementary material is available at Forestry online and includes observed net volumes for lumber and non-lumber products as a function of DBH for each species, probabilities of occurrence and predicted conditional volumes for the best models featured in this paper, results of cross-validation for the best conditional volume model and relative RMSE by species and product type as a function of the number of trees left out during the cross-validation process for the best marginal volume model. Acknowledgements We would like to thank Étienne Boulay, Jocelyn Hamel, Éric Labrecque, Pierre Laurent and Jean-François Leblond for their contributions in field measurements, Claude Jolivet and Alain Langevin for log classification, Yves Giroux, Luc Bédard, Ghislain Veilleux and Francis Tanguay from FPInnovations for log sawing, lumber grading and data compilation. We also thank SÉPAQ Duchesnay and École forestière et de Technologie du bois de Duchesnay for their collaboration on this project, Filip Havreljuk and two anonymous reviewers for their helpful comments, as well as Denise Tousignant for English editing of this paper. Funding This study was funded by the ministère des Forêts, de la Faune et des Parcs du Québec (forest research project number 142332022) and by FPInnovations and the Canadian Wood Fibre Centre of Natural Resources Canada (Hardwood Research Initiative). Conflict of interest statement None declared. References Allison , P.D. 2012 Logistic Regression Using SAS: Theory and Application . 2nd edn . SAS Institute Inc . Baral , S.K. , Schneider , R. , Pothier , D. and Berninger , F. 2013 Predicting sugar maple (Acer saccharum) discoloured wood characteristics . Can. J. For. Res. 43 , 649 – 657 . Google Scholar CrossRef Search ADS Bédard , S. , Guillemette , F. , Raymond , P. , Tremblay , S. , Larouche , C. and DeBlois , J. 2014 Rehabilitation of northern hardwood stands using multicohort silvicultural scenarios in Québec . J. For. 112 , 276 – 286 . Burnham , K.P. and Anderson , D.R. 2002 Model Selection and Multimodel Inference: A Practical Information—Theoretic Approach . 2nd edn . Springer , p. 488 . Burton , J.I. , Zenner , E.K. and Frelich , L.E. 2008 Frost crack incidence in northern hardwood forests of the southern boreal-north temperate transition zone . North. J. Appl. For. 25 , 133 – 138 . Cockwell , M. and Caspersen , J.P. 2014 Sources of variation in the net value of sugar maple trees: implications for tree selection and operations management . For. Prod. J. 64 , 250 – 258 . Cecil-Cockwell , M.J.L. and Caspersen , J.P. 2015 A simple system for classifying sugar maple vigour and quality . Can. J. For. Res. 45 , 900 – 909 . doi:10.1139/cjfr-2014-0469 . Google Scholar CrossRef Search ADS Cunningham , R.B. and Lindenmayer , D.B. 2005 Modeling count data of rare species: some statistical issues . Ecology 86 , 1135 – 1142 . doi:10.1890/04-0589 . Google Scholar CrossRef Search ADS Drouin , M. , Beauregard , R. and Duchesne , I. 2009 Between tree variability of wood color in paper birch (Betula papyrifera Marsh.) in Québec . Wood Fiber Sci. 41 , 333 – 345 . Drouin , M. , Beauregard , R. and Duchesne , I. 2010 Impact of paper birch (Betula papyrifera) tree characteristics on lumber color, grade recovery, and lumber value . For. Prod. J. 60 , 236 – 243 . Duchesne , I. , Vincent , M. , Wang , X. , Ung , C.-H. and Swift , E. 2016 Wood mechanical properties and discoloured heartwood proportion in sugar maple and yellow birch grown in New Brunswick . BioResources 11 , 2007 – 2019 . Google Scholar CrossRef Search ADS Firth , D. 1993 Bias reduction of maximum likelihood estimates . Biometrika. 80 , 27 – 38 . Google Scholar CrossRef Search ADS Fortin , M. , Guillemette , F. and Bédard , S. 2009 Predicting volumes by log grades in standing sugar maple and yellow birch trees in southern Québec, Canada . Can. J. For. Res. 39 , 1928 – 1938 . Google Scholar CrossRef Search ADS FPInnovations . 2014 . Optitek 10: user’s manual. FPInnovations, Québec, Canada. Germain , R.H. , Yanai , R.D. , Mishler , A.K. , Yang , Y. and Park , B.B. 2015 Landscape and individual tree predictors of dark heart size in sugar maple . J. Forestry 113 , 20 – 29 . Google Scholar CrossRef Search ADS Gong , M. , Tu , D. , Li , L. and Chui , Y.H. 2015 . Planar shear properties of hardwood cross layer in hybrid cross laminated timber. Proceedings of the 5th International Scientific Conference on Hardwood Processing (ISCHP2015), Sept. 15–17, Québec, Canada. Gregoire , T.G. , Lin , Q.F. , Boudreau , J. and Nelson , R. 2008 Regression estimation following the square-root transformation of the response . For. Sci. 54 , 597 – 606 . Guillemette , F. , Bédard , S. and Fortin , M. 2008 Evaluation of a tree classification system in relation to mortality risk in Québec northern hardwoods . For. Chron. 84 , 886 – 899 . Google Scholar CrossRef Search ADS Guillemette , F . 2016 . Diamètres à maturité pour l’érable à sucre et le bouleau jaune au Québec. Gouvernement du Québec. Ministère des Forêts, de la Faune et des Parcs, Direction de la recherche forestière. Note de recherche forestière no. 145. 14 p. Hanks , L.F. 1976 a. Hardwood tree grades for factory lumber. USDA For. Serv. Northeast. For. Exp. Stn. Res. Pap. NE-333. 81 pp. Hanks , L.F. 1976 b. How to predict lumber-grade yields for graded trees. USDA For. Serv. Northeast. For. Exp. Stn. Gen. Tech. Rep. NE-20. 9 pp. Havreljuk , F. , Achim , A. , Auty , D. , Bédard , S. and Pothier , D. 2014 Integrating standing value estimations into tree marking guidelines to meet wood supply objectives . Can. J. For. Res. 44 , 750 – 759 . Google Scholar CrossRef Search ADS Havreljuk , F. , Bédard , S. , Guillemette , F. and DeBlois , J. 2015 . Predicting log grade volumes in northern hardwood stands in southern Québec. Proceedings of the 5th International Scientific Conference on Hardwood Processing (ISCHP2015), Sept. 15–17, 2015, Québec, Canada. Hosmer , D.W. and Lemeshow , S. 2000 Applied Logistic Regression . 2nd edn . John Wiley & Sons, Inc . Google Scholar CrossRef Search ADS Kubler , H. 1983 Mechanisms of frost crack formation in trees—a review and synthesis . For. Sci. 29 , 559 – 568 . Leak , W.B. , Yamasaki , M. and Holleran , R. 2014 . Silvicultural guide for northern hardwoods in the northeast. United States department of Agriculture, Forest Service, General Technical report NRS-132, Northern research station, 46 pp. Majcen , Z. , Richard , Y. , Ménard , M. and Grenier , Y. 1990 . Choix des tiges à marquer pour le jardinage d’érablières inéquiennes. Guide technique. Ministère de l’Énergie et des Ressources du Québec. Direction de la recherche forestière. Mémoire no 96. 96 pp. https://www.mffp.gouv.qc.ca/publications/forets/connaissances/recherche/Divers/Memoire96.pdf (accessed on 19 September, 2017). MFFP (Ministère des Forêts, de la Faune et des Parcs) . 2016 . Placettes échantillons permanentes. Normes techniques. Gouv. du Québec, Ministère des Forêts, de la Faune et des Parcs, Direction des inventaires forestiers, 238 pp. https://mffp.gouv.qc.ca/publications/forets/connaissances/Norme-PEP.pdf (accessed on 19 September, 2017). MFFP (Ministère des Forêts, de la Faune et des Parcs) . 2015 . Ressources et industries forestières. Portrait statistique, édition 2015. Gouv. du Québec, Ministère des Forêts, de la Faune et des Parcs, 91 pp. www.mffp.gouv.qc.ca/forets/connaissances/connaissances-statistiques.jsp (accessed on 19 September, 2017). MRN (Ministère des Ressources naturelles) . 1995 . Classification des tiges d’essences feuillues. Normes techniques. Ministère des Ressources naturelles du Québec, Service des inventaires forestiers. 73 p. NHLA (National Hardwood Lumber Association) 2007 Rules for the Measurement and Inspection of Hardwood and Cypress Lumber . National Hardwood Lumber Association , p. 106 . OMNRF (Ontario Ministry of Natural Resources and Forestry) 2015 Forest Management Guide to Silviculture in the Great Lakes-St. Lawrence and Boreal Forests of Ontario . Queens Printer for Ontario , 394 . https://dr6j45jk9xcmk.cloudfront.net/documents/4125/revised-silvguide-mar-2015-aoda-compliant.pdf (accessed on 19 September, 2017). OMNR (Ontario Ministry of Natural Resources) . 2004 . Ontario Tree Marking Guide, Version 1.1 Ont. Min. Nat. Resour. Queen’s Printer for Ontario. Toronto, 252 pp. Pelletier , G. , Landry , D. and Girouard , M. 2013 A Tree Classification System for New Brunswick . Northern Hardwoods Research Institute , 54 pp. Petro , F.J. 1971 Felling and bucking hardwoods. How to improve your profit . Can. For. Serv., Dep. Fish. and For., Publ 1291 , 140 . Petro , F.J. and Calvert , W.W. 1976 . How to grade hardwood logs for factory lumber. Canadian Forestry Service, Department of Fisheries and the Environment, Ottawa, Ont. Forestry Technical Report 6. 67 pp. Pothier , D. , Fortin , M. , Auty , D. , Delisle-Boulianne , S. , Gagné , L.-V. and Achim , A. 2013 Improving tree selection for partial cutting through joint probability modelling of tree vigor and quality . Can. J. For. Res. 43 , 288 – 298 . Google Scholar CrossRef Search ADS Rast , E.D. , Sonderman , D.L. and Gammon , G.L. 1973 . A guide to hardwood log grading. Revised edition. USDA For. Serv., Upper Darby, PA. Gen. Tech. Rep. NE-1. 34 pp. Régnière , J. , Saint-Amant , R. and Béchard , A. 2012 . BioSIM 10: User’s manual. Can. For. Serv., Lau. For. Res. Cent., Québec, QC. Inf. Rep. LAU-X-129. 70 pp. Robitaille , A. and Saucier , J.P. 1998 . Paysages régionaux du Québec méridional. Gouvernement du Québec, ministère des Ressources naturelles, Les Publications du Québec. 213 pp. SAS Institute Inc 2013 SAS 9.4 Online Documentation . SAS Institute Inc , Available from: http://support.sas.com/documentation/94/index.html (accessed on 19 September, 2017). Trudelle , M. , Gélinas , N. and Beauregard , R. 2009 Estimation des retombées économiques directes engendrées par le réseau de création de valeur de la filière bois de feuillus durs au Québec . For. Chron. 85 , 538 – 547 . Google Scholar CrossRef Search ADS USDAFS. (United States Department of Agriculture, Forest Service) . 2012 . Forest Inventory and Analysis: field data collection procedures for phase 2 plots. Version 6.0. United States department of agriculture, Forest Service, Northern research station, Online at: http://www.fia.fs.fed.us/library/field-guides-methods-proc/docs/Complete%20FG%20Document/NRS%20FG%206.0-Oct%202012-Complete%20Document-opt.pdf (accessed on 19 September, 2017). Weiskittel , A.R. , Hann , D.W. , Kershaw , J.A. , Jr. and Vanclay , J.K. 2011 Forest Growth and Yield Modeling . Wiley-Blackwell , Oxford, 415 pp. Google Scholar CrossRef Search ADS © Institute of Chartered Foresters, 2017. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forestry: An International Journal Of Forest Research Oxford University Press

Predicting volume distributions of hardwood sawn products by tree grade in eastern Canada

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Abstract

Abstract Northern hardwoods are an ecologically and economically important forest type in eastern North America. Historically, the hardwood supply came from old-growth forests dominated by large-diameter trees. Unfortunately, the repeated removal of high-quality trees has substantially degraded hardwood forests and reduced the profitability of the primary manufacturing sector. In this context, forest managers need tools to guide silvicultural investment decisions and to estimate pre-harvest stand value based on forest inventories. The objective of this study was to evaluate the performance of classification systems and measured variables used at the tree level to predict sawn product volumes of sugar maple (Acer saccharum Marsh.), yellow birch (Betula alleghaniensis Britton) and American beech (Fagus grandifolia Ehrh.). We developed statistical models to estimate the volume of lumber products, pulpwood, sawdust and residues based on tree DBH, species, tree grades in different combinations and tree height. Results show that the tree grade variable increased the explained variation in product volumes. As expected, the accuracy of product volumes estimation, based on root mean square error (RMSE), was poor for an individual tree, but improved as the number of trees increased. Introduction The northern hardwood forest is a major forest type in eastern Canada. In Québec, it represents 16.3 per cent of the total productive forest landbase and 22.5 per cent of the total gross merchantable volume (MFFP, 2015). The province’s hardwood sawmills generate direct economic outcomes similar to the softwood industry, though at a smaller production scale (Trudelle et al., 2009). High-grade lumber is mainly used for appearance-based products such as furniture, flooring, cabinets and moudlings, while lower grades are used in packaging products (e.g. pallets). Lower-grade hardwoods can also be used for new structural applications (e.g. in cross-laminated timber for non-residential multistory buildings; Gong et al., 2015). Historically in eastern Canada, the hardwood supply came from old-growth forests where large-diameter trees were abundant. Despite limited knowledge about the resource and with only simple processing technologies, it was relatively easy to extract high-value products from those trees. However, over the years, the repeated removal of high-quality trees has substantially degraded hardwood forests and reduced the industry’s profitability to marginal levels. In this context, provincial forest managers need detailed knowledge of the current quality and value of hardwood forests to justify silvicultural investments and wood allocation. They need tools or models that can estimate the quantity and quality of products at the tree or stand level (i.e. enhanced forest inventory and tree marking processes that lead to a better estimate of pre-harvest stand value). In the past few decades, several tree classification systems have been developed in the United States (USA; Hanks, 1976a), and eastern Canada (Majcen et al., 1990; MRN, 1995; OMNR, 2004; Pelletier et al., 2013). These systems can be divided into two main groups: (1) tree grading (TG) classification systems, which aim to evaluate the tree’s potential to produce lumber products (e.g. Hanks, 1976a; MRN, 1995) and (2) hybrid Risk-Product (RP) classification systems, which comprise components both for tree mortality risk and for the stem’s potential product (e.g. Majcen et al., 1990; OMNR, 2004). Tree grading systems are generally used in national forest inventory programs to evaluate standing wood quality (USDAFS, 2012; MFFP, 2016). Hybrid tree classification systems were mainly developed for pre-harvest inventories and for tree marking. They aim to evaluate a tree’s potential for lumber products and its harvest priority. Tree grading evaluation is generally too time-consuming to be a standard part of pre-harvest inventories. In eastern Canada, some models were developed to better estimate the sawlog potential of standing trees (Fortin et al., 2009; Havreljuk et al., 2015). These models estimate the net volume that a tree can generate in each log grade, but do not allow a direct volume estimation of lumber products by visual grade. Other research has also examined how net product value per tree varies within a range of defects or tree grades but they did not develop models to estimate the volume of each lumber product in a tree (Cockwell and Caspersen, 2014; Havreljuk et al., 2014; Cecil-Cockwell and Caspersen, 2015). To determine the potential value of a standing tree, we need to estimate a tree’s potential to produce lumber in current markets, which generally consider lumber by quality grades (NHLA, 2007) using a price list (e.g. Hardwood Market Report: www.hmr.com). To our knowledge, only Hanks (1976b) has developed lumber models at the tree level based on tree grades in the USA. However, we hypothesized that these may not be applicable in stands located near the northern edge of their climatic range in eastern Canada. In this region, trees are exposed to extreme climatic conditions that promote stem defects such as frost cracks, which in turn reduce lumber value (Kubler, 1983; Burton et al., 2008). This study evaluated the performance of two tree classification systems to predict the volume of lumber grades, pulpwood, sawdust and residues for the three most important tree species found in northern hardwood forests: sugar maple (Acer saccharum Marsh.), yellow birch (Betula alleghaniensis Britton) and American beech (Fagus grandifolia Ehrh.). By predicting product volumes, our proposed approach allows users to adjust the monetary value of any product for a given market and obtain an assessment of current stand value. We also tested whether tree classes or tree grades could be merged to simplify the systems and make the estimations less time-consuming. Methods Study site The study was carried out in the Duchesnay Forest (lat. 46° 57′ N, long. 71° 40′ W) near Québec City, Canada. This forest is located in the meridional subregion of the balsam fir (Abies balsamea [L.] Mill.)–yellow birch bioclimatic domain (Robitaille and Saucier, 1998). Depending on elevation and aspect, the vegetation is characterized either by yellow birch–balsam fir stands or by sugar maple–yellow birch stands. Stands selected for this study are dominated by yellow birch, sugar maple and American beech (which, respectively, represent 38 per cent, 34 per cent and 23 per cent of merchantable basal area, i.e. that of trees with a diameter at breast height [DBH] larger than 9 cm). Podzolic soils in the region developed from deep glacial till. The bedrock is mainly composed of granitic gneiss from the Grenville Province. According to the climate estimations generated by BioSIM for the 1980–2009 period (Régnière et al., 2012), the mean annual temperature, and the mean annual maximum and minimum temperatures at the study site, are 3.0°C, 30.4°C and −35.3°C, respectively. Sampling and tree assessment A total of 52 yellow birch, 52 sugar maple and 35 American beech trees were sampled mostly from those marked for cutting in eight experimental units (1.2 ha) distributed within two different blocks of a study set up to compare different rehabilitation silvicultural treatments (Bédard et al., 2014). In some of these units, especially for the highest grades, it was necessary to select unmarked trees in the buffer zone surrounding the experimental units (20 m around the central plot) in order to cover both the desired range of DBH (i.e. 24–70 cm) and tree grades. Despite this, certain tree grades were very scarce and yielded smaller samples than others (e.g. grade A, Table 1). Total tree height and merchantable tree height (taken at the base of the crown’s main branches) were also estimated using a Vertex III (Haglöf, Sweden AB). Mean total tree height (±1 standard deviation) was 22.5 ± 3.2 m for yellow birch, 21.1 ± 3.0 m for sugar maple and 21.7 ± 2.8 m for American beech; mean merchantable tree height for these species was 13.0 ± 2.5 m, 13.2 ± 2.0 m and 13.0 ± 2.1 m, respectively. Before felling, trees were numbered, and species and DBH were recorded. Each tree was also classified according to the two classification systems described below. Table 1 Number of trees harvested, mean, standard deviation and range of tree DBH, by species and tree classification system. Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Note: n = number of trees harvested for each tree classification group for every species; SD = standard deviation; TG = tree grade classification, grade A being the highest quality and grade D, the lowest; RP = risk-product classification; AGS = low mortality risk, sawlog potential; UGS = high-mortality risk, sawlog potential; Cull = low or high-mortality risk, pulpwood or firewood potential. Table 1 Number of trees harvested, mean, standard deviation and range of tree DBH, by species and tree classification system. Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Tree classification system/class Yellow birch Sugar maple American beech n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) n Mean (cm) SD (cm) Range (cm) TG  A 5 50.2 5.0 43.5–57.3 5 45.3 6.8 40.1–53.4 0  B 12 47.6 12.0 34.5–68.2 14 38.3 4.6 33.1–49.8 5 42.1 6.5 36.9–52.5  C 18 37.7 9.8 24.6–62.4 19 40.2 10.6 26.5–73.6 15 32.7 7.8 23.1–46.3  D 17 33.8 5.6 24.4–41.1 14 37.5 5.4 25.6–46.7 15 35.4 8.7 25.2–50.2 RP  AGS 19 40.3 11.5 24.4–62.4 19 39.3 5.8 31.3–53.4 12 35.4 8.5 23.1–50.2  UGS 24 41.1 11.6 25.6–68.2 24 39.9 9.7 26.5–73.6 12 38.8 8.9 23.6–52.5  Cull 9 36.1 2.8 31.7–39.6 9 38.6 6.7 25.6–46.7 11 31.2 6.3 25.2–46.7  Total 52 39.9 10.6 24.4–68.2 52 39.5 7.8 25.6–73.6 35 35.2 8.4 23.1–52.5 Note: n = number of trees harvested for each tree classification group for every species; SD = standard deviation; TG = tree grade classification, grade A being the highest quality and grade D, the lowest; RP = risk-product classification; AGS = low mortality risk, sawlog potential; UGS = high-mortality risk, sawlog potential; Cull = low or high-mortality risk, pulpwood or firewood potential. The TG classification system separates trees into four grades (A, B, C and D) according to their DBH and their external defects on the best 3.7-m stem section within the lower 5 m (Figure 1). Grade A stems represent the best quality, i.e. trees with the highest sawlog potential in the lower 5-m section, while grade D stems have no sawlog potential. Trees must meet a specific minimal DBH for each grade (A: > 39 cm, B: > 33 cm, C and D: > 23 cm). External defects are assessed in order to estimate the length of the defect-free sections on each of the four faces. The yield of the third-best face and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade (Figure 1). The defect-free section must total at least 3.1, 2.5 and 1.8 m in length for grades A, B and C, respectively. It could be in one or two sections of at least 1.5 m in length for grade A, in one to three sections of at least 1 m in length for grade B, and in one to three sections of at least 0.6 m in length for grade C. Grades A, B and C denote trees with potential sawlogs and are similar to tree grades 1, 2 and 3 used by Hanks (1976a) in the USA. Grade D is equivalent to the ‘below grade’ in Hanks’ classification, and indicates possible local use of bolts or pulpwood. The main differences between tree grades used in Québec and in the USA are the conversion from imperial to metric units and the absence of a minimum diameter inside bark at the top of the grading section. Figure 1 View largeDownload slide Standing tree classification system based on the best 3.7-m stem section within the lower 5 m. External defects are assessed in order to estimate the length of the defect-free sections on each four faces (tree circumference is divided into four equal parts). The yield of the third-best face, tree DBH and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade. Figure 1 View largeDownload slide Standing tree classification system based on the best 3.7-m stem section within the lower 5 m. External defects are assessed in order to estimate the length of the defect-free sections on each four faces (tree circumference is divided into four equal parts). The yield of the third-best face, tree DBH and the percentage of volume reduction for cull, sweep, rot and other defects within the butt log determine the final tree grade. The hybrid RP classification system encompasses two components: tree mortality risk and a stem’s potential product. Each component is assessed using a two-level classification developed for tree marking in selection cuttings (Majcen et al., 1990). The RP classification aims to identify trees that are at high risk of mortality or of losing stem quality before the next cutting cycle (i.e. 15–25 years). Basically, it is based on crown, root and bole defects. If more than one-third of the crown shows evidence of dieback or damage, or if some major defect can be observed (e.g. presence of fungi, canker, decay, major root damage or large open wounds), the tree is assigned to the high-mortality risk class. Otherwise, it is considered as having a low risk of mortality (see Guillemette et al., 2008 for more details). Likewise, depending on bole straightness and external defects, hardwood trees are classified as pulpwood or potential sawlogs. The minimum requirement for potential sawlogs was the presence of a 1.8-m log, at any location on the stem, with one face free of defects and with no deduction for rot applicable to this grading section. The RP classification comprises four classes: (1) low mortality risk–sawlog potential, (2) low mortality risk–pulpwood or firewood, (3) high-mortality risk–sawlog potential and (4) high-mortality risk–pulpwood or firewood. However, these classes were grouped into three categories: (i) acceptable growing stock (AGS), which corresponds to the first class, (ii) unacceptable growing stock (UGS), equivalent to the primary class 3 and (iii) cull, composed of classes 2 and 4. These categories are often used in pre-harvest forest inventories, in marking guides and for silvicultural prescriptions in hardwood forests of eastern North America (OMNR, 2004; Leak et al., 2014; OMNRF, 2015). Bucking and sawmilling The selected trees were felled and topped by a harvester according to a whole-tree harvesting system and brought to the forestry school lumber mill in Duchesnay, Québec. After visual assessment of external characteristics, each stem was bucked into logs following the procedure described by Petro (1971) in order to maximize the production of high-grade lumber products (NHLA, 2007). Each log was then graded (F1, F2, F3, from the highest to the lowest quality) for factory lumber (Rast et al., 1973; Petro and Calvert, 1976), local-use bolt (F4) or pulpwood (P). The main grading factors for sawlogs are their position in the stem, length, small-end diameter, end defects, length and number of defect-free sections (clear cuttings), and scaling deductions for sweep, crook or rot. While the smallest sawlog must be at least 2.5 m long with a diameter of 20 cm at the small end, the local-use bolt grade accepts lengths down to 1.9 m and diameters down to 16 cm. The minimum requirement for pulpwood was a 2.44-m long, relatively straight piece with a small-end diameter (SED) of at least 9.1 cm. Some cull pieces, for which conversion to products was impractical (Rast et al., 1973), were classified as residues after bucking. The 139 sample trees were cut into 408 logs, of which 139 were classified as sawlogs. The 139 sawlogs were sawn into 1762 boards (yellow birch: 818; sugar maple: 656; American beech: 288) and 128 central blocks (yellow birch: 54; sugar maple: 52; American beech: 22), using a sawing-around cutting pattern. This process consists in feeding a carriage with a log, aligning the log against a saw blade to open the best possible face, extracting one or more boards, and then rotating the log at 90° before extracting another board. This operation continues until a central block (pallet cant) of lower visual quality is obtained. The number of rotations around the log depends on its size, form, internal colouration and defects. This selective sawing aims to maximize the production of defect-free boards in the high value, light-coloured sapwood part of the log. Sawn boards were 1.22–3.66 m (4–12 ft) long, 7.6–33.0 cm (3–13 inches) wide and 4/4 (25.4 mm or 1 inch) thick. In reality, the sawmill processed each board with an overthickness of 2.54 mm (0.1 inch). This common practice allows for product finishing/surfacing (e.g. planing, sanding). Thus, the actual board thickness used to calculate lumber volumes in this study was 27.94 mm (1.1 inches). Boards were graded in the rough green state under the National Hardwood Lumber Association’s grading rules (NHLA, 2007), which are based on the percentage of clear defect-free wood on a board. Central blocks (dimension: 4 × 4 inches, sawn without overthickness) were not further transformed into pallet wood at the sawmill. A complete tree–log–lumber tracking was kept throughout the study. The volume of all logs was calculated with Smalian’s formula, using the diameter and length data recorded at bucking. Lumber products were compiled in number, volume and quality for each log and each tree. Lumber volume (length × width × thickness) was calculated in imperial units and then converted into metric units (m3) to be consistent with the other tree product volumes. Sawdust volume was estimated as 15 per cent of the sawlog volume (average value of all sawlogs) based on Optitek sawing simulations (FPInnovations, 2014). Overall pulpwood volume per tree was calculated by adding the volume of logs classified as pulpwood at bucking (at least 2.44 m long and 9.1 cm in SED) to the volume of pulpwood produced at the sawmill (i.e. slabs and poor quality lumber below NHLA grades). The volume of ‘sawmill-generated’ pulpwood per log was estimated by subtracting lumber and sawdust volumes from the gross sawlog volume. When a sawlog contained decay, the estimated volume of decayed wood was added to the sawdust volume previously estimated using Optitek, since pulp mills usually do not want decayed pulpwood (chips) in their supply. Model development The models’ dependent variables correspond to the net volume (m3) of each product within a tree. Nine different product types were retained, the first five being groups of NHLA grades (NHLA, 2007) defined as Select (select sap, select red, select), No. 1 Common (1Com), No. 2 Common (2Com), No. 3 A Common (3ACom) and No. 3B Common (pallet, 3BCom); these were followed by blocks, pulpwood, residues and sawdust. Table 2 presents descriptive statistics for net volumes by species and product type and Figure S1 in the Supplementary Material shows the observed net volumes for lumber and non-lumber products as a function of DBH for each species. Table 2 Number of trees that produced a final product, mean, standard deviation and range of tree volume, by species and product type. Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Note: n = number of trees of each species that produced volume of each product; SD = standard deviation; Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3 A Common; 3BCom = No. 3B Common (pallet). All trees produced pulpwood while some trees did not produce any lumber products. Table 2 Number of trees that produced a final product, mean, standard deviation and range of tree volume, by species and product type. Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Product type Yellow birch Sugar maple American beech n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) n Mean (m3) SD (m3) Range (m3) Select 28 0.150 0.112 0.008–0.489 26 0.067 0.053 0.005–0.192 9 0.089 0.085 0.020–0.266 1Com 26 0.068 0.063 0.006–0.207 30 0.061 0.038 0.011–0.177 13 0.045 0.045 0.003–0.152 2Com 30 0.054 0.044 0.005–0.183 31 0.048 0.039 0.004–0.200 13 0.049 0.041 0.005–0.137 3ACom 25 0.029 0.019 0.003–0.070 27 0.036 0.022 0.007–0.094 13 0.031 0.017 0.013–0.074 3BCom 20 0.011 0.010 0.003–0.047 26 0.019 0.018 0.005–0.092 10 0.018 0.010 0.003–0.033 Blocks 29 0.047 0.024 0.019–0.107 31 0.045 0.020 0.013–0.082 13 0.045 0.028 0.019–0.107 Pulpwood 52 0.590 0.348 0.031–2.435 52 0.634 0.460 0.093–3.287 35 0.652 0.401 0.190–1.812 Residues 22 0.163 0.131 0.010–0.568 13 0.135 0.172 0.007–0.671 12 0.086 0.068 0.011–0.262 Sawdust 30 0.111 0.089 0.015–0.363 32 0.085 0.054 0.009–0.241 14 0.068 0.052 0.009–0.151 Note: n = number of trees of each species that produced volume of each product; SD = standard deviation; Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3 A Common; 3BCom = No. 3B Common (pallet). All trees produced pulpwood while some trees did not produce any lumber products. Since volume distributions showed an excess of zero values for most products, we used a two-part conditional model (Cunningham and Lindenmayer, 2005). The first part of the model predicted whether product j was observed in tree i. If this was the case, the volume of product j was then predicted by the second part of the model, which excluded all zero values. The final volume predictions, or marginal volumes, were then obtained by multiplying the probability of occurrence predicted by the first part and the conditional volume predicted by the second part. For the first part, the probability of observing product j in tree i ( pij) was obtained using a generalized linear model with a logit link function: log(pij1−pij)=Zijγ, (1) where Zij is a row vector of explanatory variables related to product j of tree i, and γ corresponds to the vector of unknown fixed-effects parameters. The second part of the model predicted volume vij, i.e. the volume of product j in tree i, given that vij > 0, so conditional on the presence of product j in tree i (i.e. vij|pij). The volumes were square root transformed to achieve normality and to prevent the prediction of negative volumes. The vector of within-tree square root transformed volumes ( yi=vi, with vi being the vector of vij) was assumed to follow a multivariate normal (MVN) distribution: yi~MVN(Xiβ,Ri), (2) where Xi is a matrix of independent variables for tree i, β is a vector of unknown fixed-effects parameters, and Ri is the generalized variance–covariance matrix of the within-tree error terms (UN, unstructured), used to take into account possible correlations between the products within a tree. The predicted square root transformed volumes were back-transformed to their original scale after correcting for the bias induced by the transformation (Gregoire et al., 2008): E(vi|pi)=(Xiβˆ)2+diag(Rˆi), (3) where E(vi|pi) is the mathematical expectation of vector vi, and diag(Rˆi) corresponds to the diagonal elements of Rˆi, which are the variance estimates of each product ( σˆj2). Marginal predictions of the volumes were finally obtained by multiplying the conditional predictions by the probabilities of observing the products (equation [1]): E(vij)=E(vij|pij)×pij. (4) Model specification and diagnostic The explanatory variables used to formulate the candidate models were species, DBH, total tree height (TH), merchantable tree height, TG classification (A, B, C, D) with different grade groupings, and RP classification (AGS, UGS, Cull). Interactions between these variables were also tested. For the first part of the model predicting the presence of the products, the parameters of significant variables were estimated using the maximum likelihood (ML) method in the SAS LOGISTIC procedure (SAS Institute Inc., 2013) with the logit link function. Firth’s bias correction (Firth, 1993) was used to reduce bias in the parameter estimates and to solve convergence problems. For the second part of the model, significant variables and the estimates of the final models were determined using the REML (restricted maximum likelihood) method in the SAS MIXED procedure. The various candidate models for both parts (probability of occurrence and conditional volume) were then compared and ranked using the Akaike (AIC and AICC) information criteria (Burnham and Anderson, 2002) obtained with the ML (maximum likelihood) method: the smaller the value of AIC, the better the model. The area under the receiver operating characteristic curve (AUC) (Allison, 2012) was also used to determine the best candidate model for the probability of occurrence. We considered that an AUC > 0.80 indicated that the model was accurate. The Hosmer–Lemeshow test (Hosmer and Lemeshow, 2000), for which a significant probability related to the χ2 statistic indicates a lack of fit, was also used. Normality and homogeneity of variance of the conditional volume model were checked visually with normalized (scaled) plots of residuals taking into account the autocorrelation between products, expressed by the R generalized variance–covariance matrix (equation [2]). Model evaluation The performance of each model’s predictions was evaluated at the tree level by a leave-one-out cross-validation. Every model was fitted 139 times, omitting one tree and modelling the probability of occurrence or the conditional volumes of the 138 other trees. The probabilities of occurrence and the conditional volumes of every product for the left-out tree were predicted with the resulting models. We performed a Hosmer–Lemeshow test on the predictions of the first part of the model, whereas we computed biases and root mean square errors (RMSE, as presented in Weiskittel et al., 2011) for every product and species based on predicted conditional volumes resulting from the cross-validation for the second part, after having discarded the null volumes. Moreover, we calculated biases and RMSE on marginal volume predictions obtained by multiplying the conditional volume predictions of each product of every tree by the corresponding predicted probabilities of observing the products. Finally, we calculated relative biases and relative RMSE by dividing the resulting values by the mean observed volume. We also assessed model performance for a larger number of trees based on subsets of randomly selected trees, in order to verify whether accuracy (RMSE) was improved. This was calculated from a leave-k-out cross-validation in which 5–45 trees (by increments of 5) of the 139 trees were removed from the calibration dataset at each iteration. We stopped at a maximum of 45 trees, which represented one-third of the sample trees. The remaining two-thirds or more of the sample were used to calibrate both parts of the model. Marginal volume of every product of each left-out tree was then calculated by multiplying the probability of occurrence with the corresponding conditional volume. The mean of the observed and predicted marginal volumes of every product for each species of the left-out group were calculated. We repeated this procedure 1000 times, and calculated biases and RMSE on the 1000 resulting mean values. Results Several candidate models were tested for both parts of the statistical model; the best candidate models for each combination are presented in Tables 3 and 4. Although other models that included significant interactions between tree grade and species or DBH showed smaller log-likelihood values than the best candidate models, they were not retained because the higher value of their information criteria indicated that simpler models were better. We also tested total tree height and merchantable height, alone and in interaction with other variables, but the resulting models were not better than those presented. Table 3 Model comparison of the probability of occurrence of a particular product (first part of the model) based on Akaike criteria value (AIC), area under the curve (AUC) and Hosmer and Lemeshow’s goodness-of-fit test. Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Note: Rank is the model ranking according to Akaike criteria value (AIC). Table 3 Model comparison of the probability of occurrence of a particular product (first part of the model) based on Akaike criteria value (AIC), area under the curve (AUC) and Hosmer and Lemeshow’s goodness-of-fit test. Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Model Explanatory variables AIC AUC H-L χ2 P>χ2 Rank 1 DBH, DBH2, species 1260 0.76 19.27 0.0135 6 2 DBH, DBH2, TG (A, B, C, D) 1041 0.86 14.65 0.0664 3 3 DBH, DBH2, TG (AB, C, D) 1031 0.85 8.38 0.3970 2 4 DBH, DBH2, RP (AGS, UGS, Cull) 1117 0.83 18.22 0.0196 5 5 DBH, DBH2, species, TH 1115 0.82 13.46 0.0970 4 6 DBH, DBH2, TG (AB, C, D), TH 940 0.87 18.58 0.0173 1 Note: Rank is the model ranking according to Akaike criteria value (AIC). Table 4 Model comparison of the conditional volume of a particular product (second part of the model) based on Akaike criteria values (AIC and AICC). Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Note: Rank is the model ranking according to Akaike criteria values (AIC and AICC). Table 4 Model comparison of the conditional volume of a particular product (second part of the model) based on Akaike criteria values (AIC and AICC). Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Model Explanatory variables AIC AICC Rank 1 DBH, DBH2, species −1550 −1521 4 2 DBH, DBH2, species, TG (A, B, C, D) −1603 −1561 2 3 DBH, DBH2, species, TG (AB, C, D) −1612 −1577 1 4 DBH, DBH2, species, RP (AGS, UGS, Cull) −1566 −1531 3 Note: Rank is the model ranking according to Akaike criteria values (AIC and AICC). For the first part, regarding the probability of occurrence of each product, the simplest model (Model 1) had the largest AIC value as well as the lowest AUC value, indicating that all other models were better to predict the occurrence of a particular product in a given tree. AUC values were >0.80 for all models, indicating that they all were accurate. In addition, all models had nonsignificant Hosmer–Lemeshow statistics at the 1 per cent significance level. Based on AIC values, the best models were those that included DBH and tree grade (TG, Models 2, 3 and 6). The best one was Model 6, which included DBH, species and TH. Its AIC value was 320 units lower than that of Model 1. Equation [1] is then: log(pij1−pij)=Zijγ=γ1,jDBHi+γ2,jDBHi2+γ3,jq+γ4,jTHi, (5) where pij is the probability of observing product j in tree i, Zij is a row vector of explanatory variables associated with product j in tree i, q corresponds to the tree grade index ( q=1,2,3 for grades AB [A or B], C and D, respectively). The parameter estimates of this equation and their standard errors are presented in Table 5. For example, the probability of observing the Select product for a grade A tree with a DBH of 50 cm and a total tree height of 24 m is: pSelect=e(−9.6641+0.2773*50−0.00325*502+0.2561*24)(1+e(−9.6641+0.2773*50−0.00325*502+0.2561*24))=0.9022. (6) Table 5 Parameter estimates and standard errors (in parentheses) of the best model predicting occurrence of each product, by product type (Model 6: DBH, DBH2, tree grade [AB, C, D] and total tree height [TH]). Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). γˆ1,j: Parameter estimate of each product for DBH effect. γˆ2,j: Parameter estimate of each product for DBH2 effect. γˆ3,j1 , γˆ3,j2 , γˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. γˆ4,j: Parameter estimate of each product for TH effect. Table 5 Parameter estimates and standard errors (in parentheses) of the best model predicting occurrence of each product, by product type (Model 6: DBH, DBH2, tree grade [AB, C, D] and total tree height [TH]). Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Product type DBH parameter DBH2 parameter Tree grade parameters TH parameter γˆ1,j (cm) γˆ2,j (cm) γˆ3,j1 (A or B) γˆ3,j2 (C) γˆ3,j3 (D) γˆ4,j (m) Select 0.2773 (0.1466) –0.00325 (0.00164) –9.6641 (3.5820) –11.4737 (3.5294) –12.9348 (3.6104) 0.2561 (0.0859) 1Com 0.2961 (0.1403) –0.00346 (0.00158) –9.3269 (3.4632) –10.6115 (3.3823) –12.5946 (3.4839) 0.2227 (0.0833) 2Com 0.2390 (0.1322) –0.00296 (0.00150) –7.8560 (3.2647) –8.9990 (3.1759) –10.8480 (3.2524) 0.2236 (0.0815) 3ACom 0.2108 (0.1344) –0.00246 (0.00150) –7.3363 (3.3115) –8.7042 (3.2267) –10.3794 (3.3024) 0.2013 (0.0811) 3BCom 0.2064 (0.1389) –0.00269 (0.00159) –8.4911 (3.3513) –8.9005 (3.2509) –10.3200 (3.3078) 0.2293 (0.0792) Blocks 0.1503 (0.1265) –0.00199 (0.00143) –5.0109 (3.1051) –6.3142 (3.0145) –8.0372 (3.0631) 0.1818 (0.0790) Pulpwood 0.2332 (0.3100) –0.00307 (0.00315) –3.3287 (8.5725) –3.3300 (8.1260) –3.1565 (8.0017) 0.1487 (0.2416) Residues –0.0187 (0.1347) 0.00053 (0.00154) –5.0623 (3.1882) –4.9969 (3.0837) –5.3958 (3.0954) 0.1976 (0.0780) Sawdust 0.2064 (0.1306) –0.00268 (0.00148) –7.1767 (3.2327) –8.2196 (3.1376) –10.2227 (3.2138) 0.2323 (0.0822) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). γˆ1,j: Parameter estimate of each product for DBH effect. γˆ2,j: Parameter estimate of each product for DBH2 effect. γˆ3,j1 , γˆ3,j2 , γˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. γˆ4,j: Parameter estimate of each product for TH effect. From the parameter estimates presented in Table 5, predicted probabilities of occurrence were generated. They are illustrated in Figure S2 (Supplementary Material) for two different tree heights corresponding to the first and third quantiles of tree height distribution of all sampled trees. Overall, the probability of obtaining lumber products increased with TG and TH. It also increased with DBH until 40 cm, and then decreased with larger DBH values. For non-lumber products, the probability of occurrence for pulpwood was near 100 per cent and did not vary with TG, TH or DBH. For residues, the probability increased with DBH and TH and was similar between tree grades AB and C, but lower for tree grade D. For sawdust, the probability increased with both TH and TG. It also increased with DBH until 40 cm, then decreased for larger DBH values. The Hosmer–Lemeshow’s goodness-of-fit test performed on predictions resulting from cross-validation was significant ( χ2 = 29.16 and P = 0.0003), but a visual comparison of observed and predicted probabilities showed no major problem, neither over all species nor by species (not shown). For the prediction of conditional volumes, the simplest model (Model 1) had the largest AIC and AICC values (Table 4). Therefore, all other models performed better than Model 1 to predict the conditional volume of a particular product in a given tree. Based on the Akaike criteria values, the best models were those that included DBH, species, and TG (Models 2 and 3). Model 3, which included DBH, species, and TG with grades AB, C and D, ranked best (with AIC and AICC values that were, respectively, 62 and 56 units lower than those of Model 1). Model 2, equivalent to Model 3 but without a grouping of grades A and B, ranked second. Considering Model 3 as the best one, the right-hand side of the model of equation [2] can be expressed as: Xijβ=β0,js+β1,jDBHi+β2,jDBHi2+β3,jq, (7) where Xij is a row vector of explanatory variables associated with product j in tree i, s is the species index ( s=1,2,3, with the three levels referring to yellow birch, sugar maple and American beech, respectively), and q corresponds to the tree grade index ( q=1,2,3 for the AB [A or B], C and D tree grades, respectively). The parameter estimates of this equation and their standard errors are presented in Table 6, along with the variance estimates of each product that are needed to transform volumes back to their original scale. For example, the conditional volume of Select lumber for a grade A yellow birch tree with a DBH of 50 cm is: vSelect=(−0.5349+0.02965*50−0.00023*502+0.06623)2+0.00945=0.2020m3. (8) Table 6 Parameter and variance estimates and standard errors (in parentheses) of the best conditional volume model, by product type (Model 3: species, DBH, DBH2 and tree grade [AB, C, D]). Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). Diag(Rˆi) is the diagonal of the generalized variance–covariance matrix. βˆ0,j1, βˆ0,j2, βˆ0,j3: Parameter estimates of each product for yellow birch, sugar maple and American beech, respectively. βˆ1,j: Parameter estimate of each product for DBH effect. βˆ2,j: Parameter estimate of each product for DBH2 effect. βˆ3,j1, βˆ3,j2, βˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. σˆj2: Variance estimate of each product. Table 6 Parameter and variance estimates and standard errors (in parentheses) of the best conditional volume model, by product type (Model 3: species, DBH, DBH2 and tree grade [AB, C, D]). Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Product type Species parameters DBH parameter DBH2 parameter Tree grade parameters Diag(Rˆ) βˆ0,j1 (Birch) βˆ0,j2 (Maple) βˆ0,j3 (Beech) βˆ1,j (cm) βˆ2,j (cm) βˆ3,j1 (A or B) βˆ3,j2 (C) βˆ3,j3 (D) σˆj2 Select –0.5349 (0.2413) –0.6274 (0.2443) –0.5907 (0.2392) 0.02965 (0.01102) –0.00023 (0.00013) 0.06623 (0.04160) –0.00237 (0.04185) 0 0.00945 (0.00177) 1Com –0.2294 (0.1908) –0.1950 (0.1926) –0.1863 (0.1841) 0.01083 (0.00848) –0.00004 (0.00010) 0.07907 (0.03327) 0.03466 (0.03349) 0 0.00555 (0.00101) 2Com –0.1095 (0.1519) –0.1067 (0.1540) –0.0738 (0.1479) 0.00958 (0.00691) –0.00005 (0.00008) 0.01515 (0.02690) –0.02368 (0.02685) 0 0.00458 (0.00079) 3ACom –0.0282 (0.1158) 0.0056 (0.1174) 0.0223 (0.1125) 0.00540 (0.00521) –0.00003 (0.00006) 0.02780 (0.02125) –0.01410 (0.02144) 0 0.00227 (0.00040) 3BCom 0.1072 (0.1291) 0.1484 (0.1309) 0.1553 (0.1259) –0.00394 (0.00602) 0.00007 (0.00007) 0.02033 (0.01904) 0.03398 (0.01956) 0 0.00189 (0.00040) Blocks –0.1536 (0.1004) –0.1598 (0.1017) –0.1313 (0.0968) 0.01453 (0.00455) –0.00016 (0.00005) 0.07492 (0.01768) 0.02880 (0.01786) 0 0.00191 (0.00033) Pulpwood 0.3609 (0.1854) 0.4028 (0.1896) 0.4495 (0.1837) 0.00525 (0.00872) 0.00015 (0.00010) –0.16320 (0.03639) –0.06115 (0.03197) 0 0.02448 (0.00301) Residues –0.1909 (0.2735) –0.2464 (0.2863) –0.2399 (0.2658) 0.02166 (0.01252) –0.00022 (0.00014) 0.07194 (0.05612) 0.02435 (0.05224) 0 0.01950 (0.00386) Sawdust –0.2839 (0.1433) –0.2807 (0.1452) –0.2549 (0.1387) 0.01659 (0.00651) –0.00008 (0.00007) 0.03897 (0.02576) 0.00902 (0.02565) 0 0.00427 (0.00076) Note: Select = select sap, select red and select; 1Com = No. 1 Common; 2Com = No. 2 Common; 3ACom = No. 3A Common; 3BCom = No. 3B Common (pallet). Diag(Rˆi) is the diagonal of the generalized variance–covariance matrix. βˆ0,j1, βˆ0,j2, βˆ0,j3: Parameter estimates of each product for yellow birch, sugar maple and American beech, respectively. βˆ1,j: Parameter estimate of each product for DBH effect. βˆ2,j: Parameter estimate of each product for DBH2 effect. βˆ3,j1, βˆ3,j2, βˆ3,j3: Parameter estimates of each product for tree grades A or B, C and D, respectively. σˆj2: Variance estimate of each product. From the parameter estimates presented in Table 6, we generated conditional volume predictions and then corrected them for the bias induced by transformation using the correction in equation [3]. The resulting predicted values are illustrated for each species in Figures S3–S5, for two different tree heights corresponding to the first and third quantiles of tree height distribution of each species. Overall, the predicted conditional volumes of most of the lumber grades increased with DBH and TG for each species. However, the conditional volume of blocks increased with DBH up to 40–45 cm and then decreased for larger DBH values for each TG and species. Conditional volume of the Select lumber grade was larger for yellow birch, followed by American beech and sugar maple. For other lumber grades, the predicted conditional volumes were similar. For non-lumber products, the conditional volume increased with DBH and decreased with tree quality. However, the conditional volume of pulpwood was higher for American beech than for the two other species. The conditional volume of residues and sawdust were similar between species. Regarding the leave-one-out cross-validation for the prediction of conditional volumes, relative bias values were variable, ranging from −30.0 per cent to 25.2 per cent, but mostly between −8.8 per cent and 8.0 per cent (Table S1). Note that negative bias indicates an overestimation, whereas a positive bias represents an underestimation. Regarding accuracy, relative RMSE values varied from 36.0 per cent to 124.9 per cent (Table S1). Finally, from the retained two-part model, marginal volume predictions were generated by multiplying the conditional volume predictions by the probabilities of observing the products. The resulting predicted values are illustrated in Figures 2–4, for two different tree heights corresponding to the first and third quantiles of tree height distribution of each species. Marginal lumber volume generally increased with DBH, TG and TH (Figures 2a, 3a and 4a). As expected, tree grade AB produced larger volumes of high-quality lumber products for a given height (Select, 1Com) as well as larger volumes of medium-quality (2Com), 3ACom and blocks than tree grades C and D. Among the five lumber grades, volumes of 3BCom were the lowest produced for all species, and this volume was similar between tree grades. Volume differences between the highest and the lowest lumber grades decreased along with tree grades: the largest differences between lumber grades were observed for grade AB trees and the smallest differences, for grade D trees. Predicted volume of the Select grade was greater for yellow birch than for other species. Pulpwood volume also increased with DBH, but contrary to lumber volume, it decreased as either tree grade or tree height increased (Figures 2b, 3b and 4b). Pulpwood volume was similar for yellow birch and sugar maple, but slightly greater for American beech for all tree grades. Sawdust and residue volumes increased with DBH, and also increased very slightly with tree grades. For each tree grade, sawdust and residue volumes were similar between species. Regarding the leave-one-out cross-validation, the relative biases for products were variable, ranging from −58.8 per cent to 20.7 per cent (Table 7). The relative biases were small for pulpwood for all species, as well as for yellow birch 3ACom and 2Com products, for sugar maple Select, 3ACom and residues, and for American beech residues (−6.3 per cent to 6.6 per cent). However, relative biases were rather large for other lumber grades for sugar maple and yellow birch (−40.9 per cent to 20.7 per cent). The biases obtained for American beech were generally larger than for other species. The biases obtained with 45-tree subsets were generally similar to those obtained at the tree level (Table 7). Regarding accuracy, the relative RMSE decreased as the number of trees in the subset increased, which means that volume estimates were much more accurate for larger subsets of trees (Figure S6 in Supplementary Material). For example, the relative RMSE values were much lower for the 45-tree subsets (15.3–101.5 per cent, Table 7) than at the tree level (44.9–292.9 per cent). Table 7 Results of the cross-validation (bias and root mean square errors [RMSE], both in absolute and relative values) for the best marginal volume model (i.e. Models 6 and 3) Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Note: n = Number of observations used for calculations. Mean volumes were calculated using all observed volumes, including null volumes for each product type. Relative RMSE values for different tree subsets are also presented in Figure S6. Table 7 Results of the cross-validation (bias and root mean square errors [RMSE], both in absolute and relative values) for the best marginal volume model (i.e. Models 6 and 3) Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Species/product type n Mean Leave-one-out cross-validation 1000 iterations of 45-tree subsets volume (m3) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Bias (m3) Bias (%) RMSE (m3) RMSE (%) Yellow birch 468 0.1016 0.0047 4.6 0.1149 113.1 0.0050 4.9 0.0336 33.1  Select 52 0.0809 0.0104 12.8 0.0851 105.2 0.0074 9.2 0.0255 31.5  1Com 52 0.0342 0.0033 9.6 0.0428 124.9 0.0018 5.2 0.0125 36.5  2Com 52 0.0313 0.0020 6.4 0.0314 100.5 0.0007 2.2 0.0092 29.3  3ACom 52 0.0140 0.0002 1.5 0.0128 91.1 –0.0005 –3.4 0.0039 27.7  3BCom 52 0.0042 −0.0017 −40.9 0.0079 190.7 –0.0022 –54.2 0.0032 77.3  Blocks 52 0.0260 0.0021 8.2 0.0212 81.3 0.0015 5.6 0.0061 23.4  Pulpwood 52 0.5904 0.0045 0.8 0.3063 51.9 0.0253 4.3 0.0901 15.3  Residues 52 0.0691 0.0143 20.7 0.1045 151.2 0.0061 8.8 0.0285 41.3  Sawdust 52 0.0640 0.0070 11.0 0.0575 89.9 0.0046 7.3 0.0170 26.6 Sugar maple 468 0.0969 0.0066 6.8 0.1463 151.0 0.0061 6.3 0.0409 42.2  Select 52 0.0334 0.0014 4.3 0.0404 121.2 –0.0007 –2.2 0.0127 38.0  1Com 52 0.0350 0.0034 9.6 0.0299 85.7 0.0021 5.9 0.0094 26.9  2Com 52 0.0288 0.0034 11.8 0.0308 107.1 0.0022 7.7 0.0086 29.9  3ACom 52 0.0189 0.0011 5.6 0.0207 109.3 0.0002 1.2 0.0055 28.9  3BCom 52 0.0093 0.0011 11.9 0.0148 159.5 0.0005 5.3 0.0041 43.6  Blocks 52 0.0268 0.0024 8.9 0.0211 78.9 0.0017 6.3 0.0060 22.3  Pulpwood 52 0.6336 0.0418 6.6 0.4194 66.2 0.0502 7.9 0.1166 18.4  Residues 52 0.0338 −0.0011 –3.1 0.0990 292.9 −0.0045 −13.2 0.0288 85.3  Sawdust 52 0.0525 0.0058 11.1 0.0474 90.3 0.0034 6.6 0.0136 26.0 American beech 315 0.0888 −0.0016 −1.8 0.1042 117.2 –0.0019 –2.1 0.0376 42.3  Select 35 0.0229 −0.0030 −13.0 0.0501 219.1 –0.0052 –22.8 0.0171 75.0  1Com 35 0.0166 −0.0050 −30.0 0.0342 206.1 –0.0066 –39.5 0.0126 76.1  2Com 35 0.0180 −0.0052 −29.1 0.0348 193.1 –0.0069 –38.2 0.0133 73.7  3ACom 35 0.0115 −0.0027 −23.3 0.0203 176.2 –0.0033 –28.9 0.0072 62.7  3BCom 35 0.0051 −0.0030 −58.8 0.0110 216.7 –0.0035 –69.3 0.0051 101.5  Blocks 35 0.0168 −0.0047 −27.8 0.0272 161.9 –0.0061 –36.3 0.0104 61.9  Pulpwood 35 0.6520 0.0213 3.3 0.2925 44.9 0.0313 4.8 0.1055 16.2  Residues 35 0.0295 −0.0019 −6.3 0.0581 197.0 −0.0047 −15.9 0.0197 66.9  Sawdust 35 0.0272 −0.0104 −38.2 0.0505 185.7 −0.0121 −44.5 0.0197 72.4 Note: n = Number of observations used for calculations. Mean volumes were calculated using all observed volumes, including null volumes for each product type. Relative RMSE values for different tree subsets are also presented in Figure S6. Figure 2 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing yellow birch trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 2 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing yellow birch trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 3 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing sugar maple trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 3 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing sugar maple trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 4 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing American beech trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Figure 4 View largeDownload slide Predicted marginal volumes of (a) lumber and (b) non-lumber products in standing American beech trees based on DBH (cm), DBH2, species, tree grade (TG) and total tree height (TH, in metres). Discussion From a statistical viewpoint, both tree classification systems analysed in this study improved the prediction of both the occurrence and the conditional volume of a given product compared with the simplest model (Model 1, Tables 3 and 4). Adding RP classes (Model 4, Tables 3 and 4) slightly improved occurrence and conditional volume estimates, but TG models, with TH for the first part, ranked best (Model 6 in Table 3 and Model 3 in Table 4). For better accuracy, the volume of each product type should therefore be estimated using both the TG classification and a measure of TH. However, from a practical viewpoint, it is not reasonable to measure TH on each tree of the sample plot during a pre-harvest forest inventory. It is more realistic to estimate TH using a local height-diameter relationship. Merging tree grades A and B in TG models resulted in better models (Models 3 and 6 in Table 3 and Model 3 in Table 4), but this could be due to the small sample size for grade A trees (Table 1). In any case, this situation reflects the reality of the high-graded northern hardwood forests, in which grade A trees are sparse. We also evaluated further simplifications of the two classifications by merging of grades. None were statistically better (based on AIC values) than Models 6 and 3, which contained grades AB (i.e. A and B grouped), and grades C and D (modelled separately), TH in the first part and species in the second part of the model. From a practical viewpoint, the most important advantage of using the TG classification is its capacity to discriminate stems with a better potential to produce higher-quality sawn products, compared with models which do not include tree quality assessment. This result indicates that better tree grades are not solely related to larger DBH, but also to external tree characteristics that influence internal wood product quality. As we mentioned before, biases were large for some products (Table 7). The best two-part conditional model (Model 6 for the first part of the model and Model 3 for the second part) was less accurate for predicting volumes for 3BCom, probably because of the smaller sampled volume for this grade. For American beech, biases were also large for most other grades, probably because of the smaller number of trees sampled compared to the other species. The cross-validation showed that using 45-tree subsets generally improved results for all product categories, even if the biases and RMSE values remained high for some products (Table 7). We tried to reduce bias and improve accuracy by grouping product categories (e.g. high [Select, 1Com, 2Com] and low lumber grades [3ACom, 3BCom, Blocks]), but we obtained similar results. Grouping product categories has the additional disadvantage of reducing the accuracy of tree value assessments, given the significant price differences for some products, especially in higher grades. Assigning an average price to higher grades (e.g. Select, 1Com and 2Com) appears questionable if, for example, the proportion of these grades are not known, because it could result in an under- or overestimation of total tree value. To reduce the biases of the best model, we would have needed to sample more trees for each species and tree grade. As expected, the accuracy of the estimated volume was low at the tree level. Although trees can be grouped according to species, DBH and tree grade following a forest inventory to evaluate potential products, a large portion of among-tree variability in product yields remained unexplained, resulting in a very low accuracy at the tree level (Table 7). Nonetheless, when the objective is to estimate the value of standing hardwood timber prior to cutting based on a pre-harvest forest inventory, high accuracy is not required at the individual tree level. Our simulations showed that the relative accuracy (RMSE) improved with the number of trees retained in the subsets (Figure S6, Supplementary Material). If we assume that relative accuracy would continue to improve (to some extent) for subsets of more than 45 trees, we can expect more accurate yield predictions for a larger number of trees. This hypothesis could not be tested due to the limited sample size. Moreover, we did not evaluate the accuracy at the stand level. Biases for any particular stand type or site may also exist. For example, Germain et al., 2015 found that dark heartwood size of sugar maple trees varied between sites in New York State. They found that sites with less acid soils had fewer large dark hearts, and consequently trees of higher value because high value of sugar maple lumber depends on the wood being light-coloured. The marginal lumber volumes predicted for yellow birch and sugar maple with our best two-part conditional model were much smaller than those predicted by Hanks (1976b) in the USA. The difference generally decreased with DBH for each grade. For example, for a grade 3 yellow birch stem (similar to tree grade C in our model) with a merchantable height of 13 m (43 feet), a total height of 22.5 m, our model predicted a volume for 1Com that represents 17 per cent of the volume predicted by Hanks for a DBH of 30 cm, and only 25 per cent for a DBH of 40 cm. The frequent observations of defects on the bole of our study trees, such as cracks, seams and dead branches (data not shown), combined with the common presence of a dark heartwood in sugar maples, could explain the smaller volumes of lumber grades. Compared with the study area of Hanks (1976b), ours is located at the northern limit of the distribution range of northern hardwoods, where lower stem quality can be expected. Indeed, we observed an apparently lower potential to produce quality lumber and a higher potential for pulpwood products in our study area. In a similar study in the same bioclimatic domain in which paper birch trees (Betula papyrifera Marsh.) were graded by harvesting priority based on mortality risk, Drouin et al. (2010) showed that DBH was the most important variable affecting lumber quality and value, followed by tree harvest priority. Larger trees were associated with higher lumber quality and higher lumber value in $/m3 per tree. This trend supports silvicultural treatments aimed at producing larger trees. However, the proportion of discoloured wood increases as trees grow older (Baral et al., 2013; Duchesne et al., 2016) potentially causing value loss. A more balanced approach may be needed, depending on the ability of each species to recover from traumatic events (wounds, fungal attack, etc.) that cause discoloration (i.e. to find a trade-off between tree size, discoloration and mortality risks for each species). Another study on paper birch established that larger and less vigorous trees produced boards with higher proportions of discoloured wood (Drouin et al., 2009); the effect of tree age was indirect through tree diameter. These results illustrate the complex interplay or dynamics between tree diameter growth, vigour and mortality risk caused by biological processes affecting colour and soundness of wood, which, altogether, influence timber value. It also highlights the need to have better operational tools to predict when trees have reached their maximum value (Pothier et al., 2013; Guillemette, 2016). In a value chain perspective, the poor quality of hardwood stands could make tree grading and marking economically unviable. If we assume that tree quality is sufficiently good to cover the costs of tree grading, then we think that tree grading should primarily focus on identifying only high- and low-potential trees, in order to lower costs. High-potential trees would be those with a DBH > 23 cm having at least the defect-free sections and volume reductions of grade B, since grade AB yielded much larger marginal volumes of higher grade products than other tree grades. Low-potential trees would be trees that do not qualify for grade B. This dichotomic approach would simplify tree grading and reduce the time required to grade, because only the criteria for grade B would be applied in the field. In order to evaluate actual stand value in northern conditions where grade A trees are very scarce and often hide overgrown internal defects, we could rely on evaluating only high and low grades on larger trees (i.e. DBH > 33 cm). However, in order to evaluate future stand quality, one must also consider high-potential trees in the smaller diameter classes (i.e. DBH 23–33 cm) especially in uneven-aged stands. Such crucial information on the potential quality of the forests to come (i.e. regarding the quality of tree recruitment, by grading high-potential trees) would strongly support sustainable forest management. Knowing the forest value could be more critical for decision-making in hardwood stands located at the northern limit of the species’ distribution range than on more southern sites. Indeed, in the sites studied by Hanks (1976b) in the USA, trees were more likely to produce lumber and generate revenues. The incentive for grading trees will also depend on market price differences between lumber grades, and on the demand for transforming the large volumes of pulpwood (which can represent, on average, 65 per cent of the volume of the trees analysed) associated with lumber production. The higher the premium for high-quality grades (Select, 1Com), the stronger the motivation could be to mark trees that are likely to produce these grades. Ultimately, stand value assessment and harvest decisions will depend on silvicultural and economic objectives. Though the subject needs more investigation, this study provides some useful insights. Conclusion The models developed in this study should be considered as a step toward better understanding the relationships between hardwood tree classification systems and their volume yield in sawn products. By modelling volume distributions of sawn products instead of total tree value, lumber product and pulpwood prices can be adjusted to any given market of interest to reflect current stand value. Our results indicate that sawn product volumes are better estimated after an initial assessment of stem quality. A system such as the tree grade (TG) system, possibly simplified by using only two categories (high- and low-potential trees), could help produce high-quality lumber grades and reduce the cost of forest inventories. Thus, product models such as those presented in this study could assist in decision-making, by simulating the impact of different forest management scenarios on stand value (e.g. when related to growth and yield models) and by estimating sawn product volume distributions and value in auctioned stands before harvest. However, additional sampling would be needed to test the hypothesis that the accuracy of our models improves at the stand scale and to allow the testing of site differences. Given their influence on product yields, various tree bucking procedures and log cutting patterns used in sawmills should also be compared. Supplementary data Supplementary material is available at Forestry online and includes observed net volumes for lumber and non-lumber products as a function of DBH for each species, probabilities of occurrence and predicted conditional volumes for the best models featured in this paper, results of cross-validation for the best conditional volume model and relative RMSE by species and product type as a function of the number of trees left out during the cross-validation process for the best marginal volume model. Acknowledgements We would like to thank Étienne Boulay, Jocelyn Hamel, Éric Labrecque, Pierre Laurent and Jean-François Leblond for their contributions in field measurements, Claude Jolivet and Alain Langevin for log classification, Yves Giroux, Luc Bédard, Ghislain Veilleux and Francis Tanguay from FPInnovations for log sawing, lumber grading and data compilation. We also thank SÉPAQ Duchesnay and École forestière et de Technologie du bois de Duchesnay for their collaboration on this project, Filip Havreljuk and two anonymous reviewers for their helpful comments, as well as Denise Tousignant for English editing of this paper. Funding This study was funded by the ministère des Forêts, de la Faune et des Parcs du Québec (forest research project number 142332022) and by FPInnovations and the Canadian Wood Fibre Centre of Natural Resources Canada (Hardwood Research Initiative). Conflict of interest statement None declared. References Allison , P.D. 2012 Logistic Regression Using SAS: Theory and Application . 2nd edn . SAS Institute Inc . Baral , S.K. , Schneider , R. , Pothier , D. and Berninger , F. 2013 Predicting sugar maple (Acer saccharum) discoloured wood characteristics . Can. J. For. Res. 43 , 649 – 657 . Google Scholar CrossRef Search ADS Bédard , S. , Guillemette , F. , Raymond , P. , Tremblay , S. , Larouche , C. and DeBlois , J. 2014 Rehabilitation of northern hardwood stands using multicohort silvicultural scenarios in Québec . J. For. 112 , 276 – 286 . Burnham , K.P. and Anderson , D.R. 2002 Model Selection and Multimodel Inference: A Practical Information—Theoretic Approach . 2nd edn . Springer , p. 488 . Burton , J.I. , Zenner , E.K. and Frelich , L.E. 2008 Frost crack incidence in northern hardwood forests of the southern boreal-north temperate transition zone . North. J. Appl. For. 25 , 133 – 138 . Cockwell , M. and Caspersen , J.P. 2014 Sources of variation in the net value of sugar maple trees: implications for tree selection and operations management . For. Prod. J. 64 , 250 – 258 . Cecil-Cockwell , M.J.L. and Caspersen , J.P. 2015 A simple system for classifying sugar maple vigour and quality . Can. J. For. Res. 45 , 900 – 909 . doi:10.1139/cjfr-2014-0469 . Google Scholar CrossRef Search ADS Cunningham , R.B. and Lindenmayer , D.B. 2005 Modeling count data of rare species: some statistical issues . Ecology 86 , 1135 – 1142 . doi:10.1890/04-0589 . Google Scholar CrossRef Search ADS Drouin , M. , Beauregard , R. and Duchesne , I. 2009 Between tree variability of wood color in paper birch (Betula papyrifera Marsh.) in Québec . Wood Fiber Sci. 41 , 333 – 345 . Drouin , M. , Beauregard , R. and Duchesne , I. 2010 Impact of paper birch (Betula papyrifera) tree characteristics on lumber color, grade recovery, and lumber value . For. Prod. J. 60 , 236 – 243 . Duchesne , I. , Vincent , M. , Wang , X. , Ung , C.-H. and Swift , E. 2016 Wood mechanical properties and discoloured heartwood proportion in sugar maple and yellow birch grown in New Brunswick . BioResources 11 , 2007 – 2019 . Google Scholar CrossRef Search ADS Firth , D. 1993 Bias reduction of maximum likelihood estimates . Biometrika. 80 , 27 – 38 . Google Scholar CrossRef Search ADS Fortin , M. , Guillemette , F. and Bédard , S. 2009 Predicting volumes by log grades in standing sugar maple and yellow birch trees in southern Québec, Canada . Can. J. For. Res. 39 , 1928 – 1938 . Google Scholar CrossRef Search ADS FPInnovations . 2014 . Optitek 10: user’s manual. FPInnovations, Québec, Canada. Germain , R.H. , Yanai , R.D. , Mishler , A.K. , Yang , Y. and Park , B.B. 2015 Landscape and individual tree predictors of dark heart size in sugar maple . J. Forestry 113 , 20 – 29 . Google Scholar CrossRef Search ADS Gong , M. , Tu , D. , Li , L. and Chui , Y.H. 2015 . Planar shear properties of hardwood cross layer in hybrid cross laminated timber. Proceedings of the 5th International Scientific Conference on Hardwood Processing (ISCHP2015), Sept. 15–17, Québec, Canada. Gregoire , T.G. , Lin , Q.F. , Boudreau , J. and Nelson , R. 2008 Regression estimation following the square-root transformation of the response . For. Sci. 54 , 597 – 606 . Guillemette , F. , Bédard , S. and Fortin , M. 2008 Evaluation of a tree classification system in relation to mortality risk in Québec northern hardwoods . For. Chron. 84 , 886 – 899 . Google Scholar CrossRef Search ADS Guillemette , F . 2016 . Diamètres à maturité pour l’érable à sucre et le bouleau jaune au Québec. Gouvernement du Québec. Ministère des Forêts, de la Faune et des Parcs, Direction de la recherche forestière. Note de recherche forestière no. 145. 14 p. Hanks , L.F. 1976 a. Hardwood tree grades for factory lumber. USDA For. Serv. Northeast. For. Exp. Stn. Res. Pap. NE-333. 81 pp. Hanks , L.F. 1976 b. How to predict lumber-grade yields for graded trees. USDA For. Serv. Northeast. For. Exp. Stn. Gen. Tech. Rep. NE-20. 9 pp. Havreljuk , F. , Achim , A. , Auty , D. , Bédard , S. and Pothier , D. 2014 Integrating standing value estimations into tree marking guidelines to meet wood supply objectives . Can. J. For. Res. 44 , 750 – 759 . Google Scholar CrossRef Search ADS Havreljuk , F. , Bédard , S. , Guillemette , F. and DeBlois , J. 2015 . Predicting log grade volumes in northern hardwood stands in southern Québec. Proceedings of the 5th International Scientific Conference on Hardwood Processing (ISCHP2015), Sept. 15–17, 2015, Québec, Canada. Hosmer , D.W. and Lemeshow , S. 2000 Applied Logistic Regression . 2nd edn . John Wiley & Sons, Inc . Google Scholar CrossRef Search ADS Kubler , H. 1983 Mechanisms of frost crack formation in trees—a review and synthesis . For. Sci. 29 , 559 – 568 . Leak , W.B. , Yamasaki , M. and Holleran , R. 2014 . Silvicultural guide for northern hardwoods in the northeast. United States department of Agriculture, Forest Service, General Technical report NRS-132, Northern research station, 46 pp. Majcen , Z. , Richard , Y. , Ménard , M. and Grenier , Y. 1990 . Choix des tiges à marquer pour le jardinage d’érablières inéquiennes. Guide technique. Ministère de l’Énergie et des Ressources du Québec. Direction de la recherche forestière. Mémoire no 96. 96 pp. https://www.mffp.gouv.qc.ca/publications/forets/connaissances/recherche/Divers/Memoire96.pdf (accessed on 19 September, 2017). MFFP (Ministère des Forêts, de la Faune et des Parcs) . 2016 . Placettes échantillons permanentes. Normes techniques. Gouv. du Québec, Ministère des Forêts, de la Faune et des Parcs, Direction des inventaires forestiers, 238 pp. https://mffp.gouv.qc.ca/publications/forets/connaissances/Norme-PEP.pdf (accessed on 19 September, 2017). MFFP (Ministère des Forêts, de la Faune et des Parcs) . 2015 . Ressources et industries forestières. Portrait statistique, édition 2015. Gouv. du Québec, Ministère des Forêts, de la Faune et des Parcs, 91 pp. www.mffp.gouv.qc.ca/forets/connaissances/connaissances-statistiques.jsp (accessed on 19 September, 2017). MRN (Ministère des Ressources naturelles) . 1995 . Classification des tiges d’essences feuillues. Normes techniques. Ministère des Ressources naturelles du Québec, Service des inventaires forestiers. 73 p. NHLA (National Hardwood Lumber Association) 2007 Rules for the Measurement and Inspection of Hardwood and Cypress Lumber . National Hardwood Lumber Association , p. 106 . OMNRF (Ontario Ministry of Natural Resources and Forestry) 2015 Forest Management Guide to Silviculture in the Great Lakes-St. Lawrence and Boreal Forests of Ontario . Queens Printer for Ontario , 394 . https://dr6j45jk9xcmk.cloudfront.net/documents/4125/revised-silvguide-mar-2015-aoda-compliant.pdf (accessed on 19 September, 2017). OMNR (Ontario Ministry of Natural Resources) . 2004 . Ontario Tree Marking Guide, Version 1.1 Ont. Min. Nat. Resour. Queen’s Printer for Ontario. Toronto, 252 pp. Pelletier , G. , Landry , D. and Girouard , M. 2013 A Tree Classification System for New Brunswick . Northern Hardwoods Research Institute , 54 pp. Petro , F.J. 1971 Felling and bucking hardwoods. How to improve your profit . Can. For. Serv., Dep. Fish. and For., Publ 1291 , 140 . Petro , F.J. and Calvert , W.W. 1976 . How to grade hardwood logs for factory lumber. Canadian Forestry Service, Department of Fisheries and the Environment, Ottawa, Ont. Forestry Technical Report 6. 67 pp. Pothier , D. , Fortin , M. , Auty , D. , Delisle-Boulianne , S. , Gagné , L.-V. and Achim , A. 2013 Improving tree selection for partial cutting through joint probability modelling of tree vigor and quality . Can. J. For. Res. 43 , 288 – 298 . Google Scholar CrossRef Search ADS Rast , E.D. , Sonderman , D.L. and Gammon , G.L. 1973 . A guide to hardwood log grading. Revised edition. USDA For. Serv., Upper Darby, PA. Gen. Tech. Rep. NE-1. 34 pp. Régnière , J. , Saint-Amant , R. and Béchard , A. 2012 . BioSIM 10: User’s manual. Can. For. Serv., Lau. For. Res. Cent., Québec, QC. Inf. Rep. LAU-X-129. 70 pp. Robitaille , A. and Saucier , J.P. 1998 . Paysages régionaux du Québec méridional. Gouvernement du Québec, ministère des Ressources naturelles, Les Publications du Québec. 213 pp. SAS Institute Inc 2013 SAS 9.4 Online Documentation . SAS Institute Inc , Available from: http://support.sas.com/documentation/94/index.html (accessed on 19 September, 2017). Trudelle , M. , Gélinas , N. and Beauregard , R. 2009 Estimation des retombées économiques directes engendrées par le réseau de création de valeur de la filière bois de feuillus durs au Québec . For. Chron. 85 , 538 – 547 . Google Scholar CrossRef Search ADS USDAFS. (United States Department of Agriculture, Forest Service) . 2012 . Forest Inventory and Analysis: field data collection procedures for phase 2 plots. Version 6.0. United States department of agriculture, Forest Service, Northern research station, Online at: http://www.fia.fs.fed.us/library/field-guides-methods-proc/docs/Complete%20FG%20Document/NRS%20FG%206.0-Oct%202012-Complete%20Document-opt.pdf (accessed on 19 September, 2017). Weiskittel , A.R. , Hann , D.W. , Kershaw , J.A. , Jr. and Vanclay , J.K. 2011 Forest Growth and Yield Modeling . Wiley-Blackwell , Oxford, 415 pp. Google Scholar CrossRef Search ADS © Institute of Chartered Foresters, 2017. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Forestry: An International Journal Of Forest ResearchOxford University Press

Published: Oct 11, 2017

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