Abstract A genus is a taxonomic unit that may contain one species (monotypic) or thousands. Yet counts of genera or families are used to quantify diversity where species-level data are not available. High frequencies of monotypic genera (~30% of animals) have previously been scrutinized as an artefact of human classification. To test whether Linnean taxonomy conflicts with phylogeny, we compared idealized phylogenetic systematics in silico with real-world data. We generated highly replicated, simulated phylogenies under a variety of fixed speciation/extinction rates, imposed three independent taxonomic sorting algorithms on these clades (2.65 × 108 simulated species) and compared the resulting genus size data with quality-controlled taxonomy of animal groups (2.8 × 105 species). ‘Perfect’ phylogenetic systematics arrives at similar distributions to real-world taxonomy, regardless of the taxonomic algorithm. Rapid radiations occasionally produce a large genus when speciation rates are favourable; however, small genera can arise in many different ways, from individual lineage persistence and/or extinctions creating subdivisions within a clade. The consistency of this skew distribution in simulation and real-world data, at sufficiently large samples, indicates that specific aspects of its mathematical behaviour could be developed into generalized or nomothetic principles of the global frequency distributions of higher taxa. Importantly, Linnean taxonomy is a better-than-expected reflection of underlying evolutionary patterns. birth–death process, genus, Linnean taxonomy, macroevolution, species-within-genus statistics, taxonomic rank INTRODUCTION The classification of organisms (systematics) does not always conform to their evolutionary history (phylogenetics). The identification of species pre-dates any kind of evolutionary paradigm, and indeed pre-dates any kind of science (Hopwood, 1959; Mayr, 1982), so it is reasonable for specialists to consider how to reconcile older and widely used systems of classification with tree-based thinking. Treatment of taxonomic ranks above the species level is the subject of extensive ongoing debate in the field of biological systematics and macroevolution (Hendricks et al., 2014; Giribet, Horminga & Edgecombe, 2016). Many authors suggest that species are real products of evolution, while higher-ranked groupings are arbitrary constructs (e.g., Stork et al., 2015). Meanwhile, Linnean ranked taxa, that represent nested groups of species, are accepted as biologically ‘real’ in other fields of science and beyond. Most fields of biology simply use taxonomic names to address their own questions. Taxonomic ‘surrogacy’ (using counts of families or genera to measure biodiversity) is applied where species-level identifications are not readily available (Gaston & Williams, 1993; Ricotta, Ferrari & Avena, 2002; Bertrand, Pleijel & Rouse, 2006; Heino, 2014). At small scales, environmental impact assessments of a single local ecosystem will generally yield equivalent results whether all present taxa are identified to species level or not (taxonomic sufficiency: Ellis, 1985; Timms et al., 2013). Taxonomic surrogacy is also used in synoptic study of the global fossil record, where species-level identifications may not be available because of preservational limitations. Counting the succession of fossil genera and families – not species – is the basis for the current understanding of macroevolution and global extinction patterns (Raup & Sepkoski, 1986; Lu, Yogo & Marshall, 2006; Alroy et al., 2008; Hendricks et al., 2014). A genus can contain many species or it can contain a single species. The issue of inconsistent genus size has been mooted as a major impediment to studying extinction, though it has rarely been addressed directly (Quental & Marshall, 2010). Taxonomic conventions for what constitutes sufficient distinction for a particular rank are not formally articulated but appear to differ among organismal groups (Avise & Liu, 2011). A better understanding of the diversity represented by the genus rank is important for attempts to estimate species diversity in any field that uses taxonomic surrogacy. The genus is the lowest commonly used rank among supraspecific classifications and the most widely used for taxonomic surrogacy; in this study, we focus on the genus to enable the gathering of a large empirical data set. Many groups of living animals and plants have a high frequency of monotypic genera and decreasing numbers of larger genera; this skew distribution is termed the ‘hollow curve’ and has been recognized and discussed since the early 20th century (e.g. Yule, 1925; Kendall, 1948; Holman, 1985). Such diversity patterns have many applications beyond the field of systematics itself. Early work compared the skew distributions seen in taxonomic rank and other natural patterns, such as body size and species–area curves (Yule, 1925; Anderson, 1974), though the interactions of these processes are not straightforward. Building directly on the observation that ranked taxonomic frequency distributions appear consistent, the ‘hollow curve’ pattern has been used to predict global species richness from higher-ranked taxa (Mora et al., 2011). Global taxonomic initiatives for living diversity face the same data limitations as studies of macroevolutionary trends in the fossil record: most higher-rank taxa have been discovered while a large proportion of species remain undescribed (Costello, May & Stork, 2013), and they are dependent on primary taxonomic data sets that may themselves be controversial (e.g. Bass & Richards, 2012). A demonstration that the hollow curve is an emergent property of evolutionary processes and consistent across various groups of organisms, rather than a potentially inconsistent taxonomic artefact, would thus have considerable power. This hollow curve has been repeatedly observed for almost a century, yet often considered puzzling (Yule, 1925; Holman, 1985; Aldous, 2001; Aldous, Krikun & Popovic, 2011). Some of the variability in genus size has even been attributed to taxonomic cultural factors, such as personality-driven tendencies in individual taxonomists towards ‘splitting’ or ‘lumping’ or human preferences for classification in smaller or larger groups (Fenner, Lee & Wilson, 1997; Scotland & Sanderson, 2004). Previous studies of genus size have focussed on ‘top–down’ approaches, developing simulations that accurately replicate the observed size-frequency distribution of taxonomic data sets (e.g. Yule, 1925; Maruvka et al., 2013), or compared the observed patterns with specific probability distributions (e.g. Scotland & Sanderson, 2004). Our aim in the present study is to use a ‘bottom–up’ approach, starting with species evolution and applying a perfectly objective classification, to examine whether or not the skew distribution in higher taxa is in conflict with underlying phylogenetic processes. Within a phylogeny, sister taxa are not necessarily of equivalent rank. The sister taxon of a genus may also be a genus, or it may be a species, a family or other higher taxon, or an unranked group of genera. This has raised questions about the viability of ranked taxa in a phylogenetic framework, though it is not necessarily problematic (Giribet et al., 2016). Importantly, it also means that observed patterns in established taxonomic classification are not equivalent to phylogenetic ‘imbalance’ or the relative size of nested and adjacent clades (Aldous, Krikun & Popovic, 2008). This is because the size-frequency distributions of subclades predicted a priori by birth–death processes may not be equivalent to those of taxonomic units recognized a posteriori. Species richness in living clades is controlled not by speciation alone but also by times of lineage persistence and extinction events, as these create ‘space’ within a clade, gaps that separate living species into discrete groups that may be treated as higher taxonomic entities (e.g. genera). Extinction processes are a critically important process to producing the species richness in a clade (Marshall, 2017). Extinction is inevitable over evolutionary time, and lineage loss within a clade creates discontinuities in phenotypic or genetic gradients, while accumulated branch lengths over clade evolution results in more diversity and hence more potential for generic splitting. Thus, there is only one evolutionary pathway to a large genus (a single rapid radiation), but there are many ways to create a small genus, such as a persistent, unbroken and relatively unchanging evolutionary lineage, or the extinction of other closely related species in a clade, or lineage persistence or extinction events nested within a larger clade that separate species into multiple genera. This may explain why clade size, like many natural phenomena, has a hollow curve (Yule, 1925; Strand & Panova, 2014). Literature in phylogenetics is often focussed on analysing rapid radiations and the causative explanations of their evolutionary history (e.g. Bond & Opell, 1998; Alfaro et al., 2009; Harmon & Harrison, 2015). Our goal here was to return to basic principles and examine large-scale emergent patterns in diversification, regardless of individual clade history, that could provide a more fundamental basis to identify where taxonomically defined genera may constitute genuine outliers. It is unclear to what extent these repeatedly observed skew distributions in conventional taxonomic genus size are influenced by the real evolutionary history of clades, and consequently it is unclear whether supraspecific diversity can be confidently translated to a probabilistic approximation of species diversity. That is, if a taxon is only identified to genus level, is it possible to establish a probability envelope of how many species it represents globally? To address this question, we compared empirical and simulation data to determine the range of behaviour in genus size-frequency distributions, and the variability of these distributions under different taxonomic algorithms and evolutionary rates. Consistent behaviours in ‘real-world’ taxonomy and in evolutionary simulations would indicate that generalized principles of systematics could lead to robust quantification of diversity from taxonomic surrogacy. Early work on mathematical approaches to macroevolution used birth–death models (Kendall, 1948) to explore the impact of speciation and extinction rates on patterns of cladogenesis (Rannala et al., 1998; Huelsenbeck & Lander, 2003). David Raup (1933–2015) and colleagues produced a computer program that they referred to as ‘MBL’ after a meeting in the Marine Biological Laboratory at Wood’s Hole, Massachusetts (Raup et al., 1973; Raup & Gould, 1974). Their explorations of the performance of birth–death models with this tool demonstrated the importance of the interplay of speciation and extinction rates (Sepkoski, 2012). These systems continue to provide a robust and elegant framework to explore macroevolutionary dynamics (Nee, 2006; Budd & Jackson, 2016). Tree simulation based on birth–death systems, with high replication resulting from modern computing power, is here used to assess whether or not genus size distribution in real-world taxonomic data can be reproduced using simple models. We imposed three algorithmic taxonomic classifications on large samples of simulated trees to compare a range of speciation and extinction parameters and their potential impacts on genus size trends. We also analysed a broad sampling of taxonomic data from living metazoans to assess the consistency of size-frequency patterns. The present work thus uses a ‘null model’ approach to assess the degree of disparity between deliberately idealized simulations with empirical data drawn from real historical taxonomy. This framework is designed to address the question of whether ranked groups are arbitrary, or whether they can be reconciled with underlying phylogenetic patterns, and presents a significant first step in developing a predictive approach to infer species diversity information from data with genus-level resolution. MATERIAL AND METHODS Real-world taxonomy We gathered comprehensive taxonomic data sets for a broad selection of animal groups. These data sets were selected primarily based on taxonomic completeness and global species coverage, and their acceptance and/or use by the community of relevant taxonomic experts. In each data set, taxa were treated to the same stringent quality checking. Each database was filtered to exclude fossil species where present and line checked to remove incomplete binomial epithets or false duplication due to genuine typographical errors. To facilitate comparisons across groups with potentially very different taxonomic conventions, it is necessary to impose certain a priori filters that could be applied to all the data sets. We did not include subspecies or subgenera in this analysis (following e.g. Alroy et al., 2001; Heim & Peters, 2011), because taxonomic species and genus ranks are the universal binomial epithet that are consistently available for all taxa. While all species are assigned to a genus, not all species are associated with a subgenus, and not all species are split into subspecies. Some prior studies on well-curated data sets of marine taxa ‘elevated’ subgeneric taxa to genus level (e.g. Raup, 1978). We consider such adjustments to be taxonomic revision that is the prerogative of relevant experts, and an aim of our study was to demonstrate whether the generic concept as normally expressed is comparable between groups, at least in terms of size distributions. We hence did not make any adjustments to the classification presented in the global taxonomic data sets we used here, even in the few groups where we have an appropriate level of expertise. Fossils were excluded not only to ensure consistency across different data sets but also to facilitate comparison with our simulations where all extinct species are excluded. We did not impose any further taxonomic refinement or interpretation, but where data sets recorded synonyms and reported them as such, only the valid accepted form was included in our analysis. These data sets include both monophyletic and non-monophyletic groupings. (Further, within the large non-monophyletic data set of marine invertebrates, some subgroups are incomplete because of non-marine species not included in the database.) We used these data to quantify the number of species in each valid genus for birds (Gill & Donsker, 2014), fish (Froese & Pauly, 2015), marine invertebrates (Boxshall et al., 2015), odonate insects (Schorr & Paulson, 2014), reptiles (Uetz & Hošek, 2015) and mammals (Wilson & Reeder, 2005). Model background Branching phylogenies can be modelled using ‘birth–death’ type models, and some emergent patterns can be understood from relatively simple mathematical properties that have been productively applied to macroevolutionary studies and have a long history in mathematical literature (e.g. Watson, 1875). The standard birth–death type model begins with a single parent lineage. At each iterative time step, there is a set probability that the lineage will split into two daughter lineages (a ‘birth’ with probability noted lambda, λ), go extinct (a ‘death’ with probability noted mu, μ) or persist unchanged (with probability 1 − λ − μ). The interactions of these parameters control several important properties of the descendent clade (Fig. 1). Firstly, the probability of total extinction of the descendant clade is determined by the ratio μ/λ: if the extinction rate is higher than the speciation rate, then the descendant clade will eventually go extinct; otherwise the probability of total extinction decreases as μ/λ drops. This ratio is illustrated in Figure 1 as the shades of grey in the probability space, where the black half above the diagonal μ = λ indicates inevitable total extinction. Secondly, the expected number of living descendent lineages at time t increases exponentially dependent on the difference (λ − μ) between speciation and extinction rates. This second property has been more frequently discussed in previous literature, especially in terms of the potential for rapid exponential growth of clades when the speciation rate exceeds the extinction rate (Raup, 1985). In biologically realistic scenarios, the values are near balanced (Marshall, 2017). This constraint, and the interaction of λ and μ have several interesting emergent properties. Any pair of parameters that have the same difference (λ − μ = constant) have the same (average) number of descendents in a fixed span of time (Fig. 1). Thus, if the speciation rate (λ) is lower than the extinction rate (μ), the expected number of descendent species goes to zero (λ − μ < 0), and the clade inevitably goes extinct (μ/λ > 1). If the speciation rate is much higher than the extinction rate, the population rapidly explodes into biologically unrealistic species richness. Figure 1. View largeDownload slide The probability space of birth–death models that generate simulated phylogenies, for rates of speciation (λ, horizontal axis) and extinction (μ, vertical axis), illustrating the main emergent properties of the model. The probability of eventual total extinction of the descendant clade is relative to the ratio μ/λ; the slope within this space is illustrated with varying shades of grey from guaranteed extinction (μ/λ > 1, black) to increasing probability of clade persistence (paler wedges correspond to ratios indicated on right vertical axis). The average number of living descendant species at a fixed sampling time point (t) is relative to the difference λ − μ, visualized as the negative intercept of a line with slope 1, and increases exponentially as et(λ − μ). Thus when λ − μ = 0.01, at t = 400, simulations produce an average of 55 species; a small increase to λ − μ = 0.02 would result in 3000 species per tree in the same timeframe. The parameters selected for simulations herein (coloured circles) were chosen to represent a span of model behaviours with consistent average clade size, but varying clade extinction probabilities (shades of grey in background). Figure 1. View largeDownload slide The probability space of birth–death models that generate simulated phylogenies, for rates of speciation (λ, horizontal axis) and extinction (μ, vertical axis), illustrating the main emergent properties of the model. The probability of eventual total extinction of the descendant clade is relative to the ratio μ/λ; the slope within this space is illustrated with varying shades of grey from guaranteed extinction (μ/λ > 1, black) to increasing probability of clade persistence (paler wedges correspond to ratios indicated on right vertical axis). The average number of living descendant species at a fixed sampling time point (t) is relative to the difference λ − μ, visualized as the negative intercept of a line with slope 1, and increases exponentially as et(λ − μ). Thus when λ − μ = 0.01, at t = 400, simulations produce an average of 55 species; a small increase to λ − μ = 0.02 would result in 3000 species per tree in the same timeframe. The parameters selected for simulations herein (coloured circles) were chosen to represent a span of model behaviours with consistent average clade size, but varying clade extinction probabilities (shades of grey in background). Synthetic taxonomy In the case of the present models, fixed speciation (λ) and extinction (μ) rates were used within each individual simulation in order to constrain the behaviour of the simulation. However, each individual simulation was relatively short (400 generations), so results are combined from large-scale replication. We generated synthetic trees using a fast C++ implementation of the MBL model (Raup et al., 1973; Supporting Information, Data S1). Random numbers were imported as 32-bit unsigned integers from a 100 Mb set of quantum random numbers downloaded from https://qrng.anu.edu.au (see Symul, Assad & Lam, 2011). Tree growth was initiated with one lineage at time t = 0 and iterated for 400 generations. The code was tested through comparison of 10 000 tree runs with predicted theoretical values of rates of total extinction and mean survivorship at t = 400. Observed values for both lay within 0.1% of predicted values (Supporting Information, Data S2). We set no limit on tree size (unlike Raup et al., 1973, who were constrained by available computer memory). The software interface allows readers to run these simulations and to manipulate generation time and threshold values for the taxonomic algorithms (Supporting Information, Data S1). We selected five pairs of values for the parameters λ (speciation probability at each iteration) and μ (extinction probability at each iteration) for use in this study. These were selected to give the same value of λ − μ = 0.01, and hence to provide the same value for mean number of species at t = 400 in all cases [calculated as et(λ − μ) = e4 ≈ 54.6 living species at time t = 400]. The parameter pairs were: λ = 0.015, μ = 0.005; λ = 0.025, μ = 0.015; λ = 0.055, μ = 0.045; λ = 0.125, μ = 0.115 and λ = 0.200, μ = 0.190 (Fig. 1). For each parameter pair, we generated 10 000 successful trees – that is, all trees that experienced total extinction before t = 400 were discarded and the simulation was continued until 10 000 lineages survived to t = 400. In the surviving trees, we excluded all extinct lineages and only considered the species (tips) extant at t = 400. We then imposed synthetic taxonomies to delineate species alive at the final sampling into ‘genera’. Three approaches to taxonomy were used: Relative Difference Taxonomy (RDT), Internal Depth Taxonomy (IDT) and Fixed Depth Taxonomy (FDT). All three algorithms produce only monophyletic genera, identified using different features of the internal topology of the tree (Fig. 2; Supporting Information, Fig. S2.1). Figure 2. View largeDownload slide Schematic representation of three independent taxonomic algorithms, applied to sort simulated species trees into monophyletic genus units. In Relative Distance Taxonomy, tips (species) that are relatively closer to each other than to the previous common ancestor are united in a genus. Here, the threshold is 0.5 or 50% of the relative depth. The depth between node a1 and b1 is more than 0.5 the depth from b1 to its alternate descendant. Thus, the two descendent lines from b1 are split into two genera. Internal Depth Taxonomy separates monophyletic of clades of tips wherever an internodal distance exceeds a given threshold (paraphyletic clusters are divided into monophyletic genera). Fixed Depth Taxonomy defines genera to be the monophyletic groups of descendants of nodes after a given depth threshold. Figure 2. View largeDownload slide Schematic representation of three independent taxonomic algorithms, applied to sort simulated species trees into monophyletic genus units. In Relative Distance Taxonomy, tips (species) that are relatively closer to each other than to the previous common ancestor are united in a genus. Here, the threshold is 0.5 or 50% of the relative depth. The depth between node a1 and b1 is more than 0.5 the depth from b1 to its alternate descendant. Thus, the two descendent lines from b1 are split into two genera. Internal Depth Taxonomy separates monophyletic of clades of tips wherever an internodal distance exceeds a given threshold (paraphyletic clusters are divided into monophyletic genera). Fixed Depth Taxonomy defines genera to be the monophyletic groups of descendants of nodes after a given depth threshold. Relative-difference taxonomy (RDT) makes no assumption that genera should be similar in age and implements a relatively complex set of rules, to formally articulate sorting from the general principles of phylogenetic systematics. This asserts that a genus should be a group containing those species that are relatively phylogenetically closer to each other than they are to anything outside the genus group. In our algorithm, all sister-species pairs were de facto united in a genus, along with any additional taxa that formed a clade without exceeding the relative distance threshold. Where the threshold is 0.5, this means more than doubling the phylogenetic distance between nodes. We tested the algorithm’s sensitivity to the relative distance threshold with four different values (0.3, 0.5, 0.6 and 0.75). All extant species not placed in a genus by this pairing/expansion algorithm are left as monospecific genera (Fig. 2; Supporting Information, Fig. S2.1). Internal-depth taxonomy (IDT) operates on a similar principle of relative differentness but uses an unrelated algorithm. Under IDT, a genus is a group of species lineages whose internodal distances are always less than a fixed threshold. Where a lineage persists without splitting for longer than the threshold distance, the downstream branches establish a new genus, and any paraphyletic genera are automatically split into monophyletic units. Four threshold values were tested, at 3.75, 5, 10 and 15% of total simulation time (15, 20, 40 and 60 time-iterations). Fixed-depth taxonomy (FDT) defines a genus to comprise all species diverging for less than a constant amount of time. Avise & Johns (1999), for example, suggested divergence at the interval of 2–5 Ma for contemporary species. FDT groups into one genus, all species whose most recent common ancestor occurred at or after a ‘threshold’ number of time-iterations from the end of the simulation. This threshold was tested at 3.75, 5, 10 and 15% of total simulation time (15, 20, 40 and 60 time-iterations) for this study. The approach provides a naive but easily understood taxonomy in which there is an absolute upper limit to the degree to which any two congeneric species can be separated from each other. Simulations were repeated with four different thresholds for each algorithm, thus producing 12 taxonomic schemes for each speciation/extinction rate parameter set. Our software allows sorting to be completed in parallel for the three algorithms, thus 20 simulations were performed (four threshold sets on each of five rate parameter pairs). Each simulation was run until 10 000 trees were produced. RESULTS Real-world taxonomy Size-frequency data of genus-level species richness are remarkably consistent among all sampled data sets (Fig. 3; Table 1; Supporting Information, Data S2). The largest fraction of genera in any group is monotypic genera (size = 1 species), decreasing nonlinearly in frequency with increasing genus size. The proportion of monotypic genera was around one-third of genera in all sampled groups (28–43%; Table 1). The behaviour of the non-monophyletic groups sampled (fish, marine invertebrates) did not differ from the other data sets. The same universal behaviour emerges in sufficiently large samples. The general pattern of (1) a skewed frequency distribution of genus size and (2) approximately one-third of genera being monotypic holds true in other subsampled partitions of monophyletic taxonomic orders (data not shown). Figure 3. View largeDownload slide Size frequency of genera in real-world taxonomic data: the percentage of genera containing a set number of valid nominal species, summarized from global data sets for select groups. Figure 3. View largeDownload slide Size frequency of genera in real-world taxonomic data: the percentage of genera containing a set number of valid nominal species, summarized from global data sets for select groups. Table 1. Summary information for valid, and taxonomically accepted, non-extinct species and genera compiled from comprehensive global taxonomic data sets Mammals Marine invertebrates Birds Reptiles Fish Dragonflies Total Number of species 5492 214 417 10 695 10 178 32 324 6043 279 149 Number of genera 1242 29 316 2278 1176 4914 688 39 614 Maximum genus size 173 1028 87 398 291 147 1028 Number of monotypic genera 538 10 970 903 329 1704 195 14 639 Species in monotypic genera 9.8% 5.1% 8.4% 3.2% 5.3% 3.2% 5.2% Proportion of genera monotypic 43.3% 37.4% 39.6% 28.0% 34.7% 28.3% 37.0% Mammals Marine invertebrates Birds Reptiles Fish Dragonflies Total Number of species 5492 214 417 10 695 10 178 32 324 6043 279 149 Number of genera 1242 29 316 2278 1176 4914 688 39 614 Maximum genus size 173 1028 87 398 291 147 1028 Number of monotypic genera 538 10 970 903 329 1704 195 14 639 Species in monotypic genera 9.8% 5.1% 8.4% 3.2% 5.3% 3.2% 5.2% Proportion of genera monotypic 43.3% 37.4% 39.6% 28.0% 34.7% 28.3% 37.0% View Large Table 1. Summary information for valid, and taxonomically accepted, non-extinct species and genera compiled from comprehensive global taxonomic data sets Mammals Marine invertebrates Birds Reptiles Fish Dragonflies Total Number of species 5492 214 417 10 695 10 178 32 324 6043 279 149 Number of genera 1242 29 316 2278 1176 4914 688 39 614 Maximum genus size 173 1028 87 398 291 147 1028 Number of monotypic genera 538 10 970 903 329 1704 195 14 639 Species in monotypic genera 9.8% 5.1% 8.4% 3.2% 5.3% 3.2% 5.2% Proportion of genera monotypic 43.3% 37.4% 39.6% 28.0% 34.7% 28.3% 37.0% Mammals Marine invertebrates Birds Reptiles Fish Dragonflies Total Number of species 5492 214 417 10 695 10 178 32 324 6043 279 149 Number of genera 1242 29 316 2278 1176 4914 688 39 614 Maximum genus size 173 1028 87 398 291 147 1028 Number of monotypic genera 538 10 970 903 329 1704 195 14 639 Species in monotypic genera 9.8% 5.1% 8.4% 3.2% 5.3% 3.2% 5.2% Proportion of genera monotypic 43.3% 37.4% 39.6% 28.0% 34.7% 28.3% 37.0% View Large The frequency distribution patterns among different organisms are visually similar and may be statistically equivalent. While the distributions differ slightly in terms of the proportion of monotypic taxa (the spread of values on the left side in Fig. 3), the question of relevance is whether these frequency distributions deviate significantly from each other over the whole span of genus sizes. Statistical tests to compare discrete distributions may have limited information value, but pairwise two-tailed Kruskal–Wallis tests on proportional frequencies (i.e. percentages of genera in each species-richness size for each taxonomic group) found no significant difference at α = 0.05 between any two groups (all pairwise comparisons P < 0.039), with the single exception of mammals and birds (pairwise comparison, D = 0.255, P = 0.0914; Supporting Information, Data S2). Mammalia is the smallest data set included in the analysis, and that deviation was driven by the size of the largest mammal genera. The two largest mammal genera are Myotis bats with 102 spp. and Crocidura shrews with 173 spp. (the largest bird genus, Zosterops, has 87 spp.). Data sets were compared based on percentages to accommodate the range of total size, and thus the one large mammal genus represents a larger proportion of total mammal genus diversity. Mammal genera have a broader range of species richness relative to birds, but neither of these two groups was significantly different from any other group, including the total group. Size-frequency distributions followed a similar pattern in all groups; however, the sizes of the largest genera were distinctly different. The largest marine invertebrate genera are an order of magnitude larger than other groups that we examined (Fig. 3; Table 1). Nonetheless, the proportions of monotypic genera were consistent (Table 1) and the overall frequency distributions are statistically equivalent (see above). Maximum genus size was also independent of taxonomic group and did not correlate with the number of genera or total group species richness (genera: P = 0.740, species: P = 0.780). Synthetic taxonomy The real-world taxonomic data (Fig. 3) and all three taxonomic rule sets (RDT, IDT and FDT) in simulation consistently recovered broadly hollow curve distributions of genus size, with proportionally higher numbers of small genera and smaller numbers of large genera (Fig. 4). In summative simulation data (combining heterogeneous speciation and extinction rates), the distributions are strongly similar to real-world data, and the proportion of monotypic genera is equivalent to that in real-world taxonomy (Fig. 4D). Simulations, however, recovered maximum genus sizes that were substantially smaller than some reported from organismal taxonomy. Figure 4. View largeDownload slide Size frequency of genera in synthetic taxonomy derived from simulated data, using five parameter sets for rates of speciation (λ) and extinction (μ), shown in different colours; the size-frequency distribution of the total ‘real-world’ data set is included for comparison (summed from data shown in Fig. 3). In each panel, solid and dotted lines indicate different thresholds for the algorithms that define synthetic genera. A, genera defined by Relative Difference Taxonomy, with a threshold of 50% difference in depth (dotted lines) or 60% (solid lines). B, genera defined by Internal Depth Taxonomy: defined by monophyletic clades of tips (species) within 20 generations (5% of tree depth, solid lines) or 40 generations (dotted lines) from any adjacent tips. C, genera defined by Fixed Depth Taxonomy: defined by monophyletic clades of tips (species) within 20 generations (5% of tree depth, solid lines) or 40 generations (dotted lines) from the most recent common ancestor. D, frequency distributions for each algorithm, summed over all speciation and extinction rate parameters (showing six different data sets from simulations: grey and real-world taxonomic data: black; symbols, dotted lines and dashed lines correspond to algorithm thresholds as in other parts). Figure 4. View largeDownload slide Size frequency of genera in synthetic taxonomy derived from simulated data, using five parameter sets for rates of speciation (λ) and extinction (μ), shown in different colours; the size-frequency distribution of the total ‘real-world’ data set is included for comparison (summed from data shown in Fig. 3). In each panel, solid and dotted lines indicate different thresholds for the algorithms that define synthetic genera. A, genera defined by Relative Difference Taxonomy, with a threshold of 50% difference in depth (dotted lines) or 60% (solid lines). B, genera defined by Internal Depth Taxonomy: defined by monophyletic clades of tips (species) within 20 generations (5% of tree depth, solid lines) or 40 generations (dotted lines) from any adjacent tips. C, genera defined by Fixed Depth Taxonomy: defined by monophyletic clades of tips (species) within 20 generations (5% of tree depth, solid lines) or 40 generations (dotted lines) from the most recent common ancestor. D, frequency distributions for each algorithm, summed over all speciation and extinction rate parameters (showing six different data sets from simulations: grey and real-world taxonomic data: black; symbols, dotted lines and dashed lines correspond to algorithm thresholds as in other parts). To exclude the possibility that maximum genus size was constrained primarily by clade size, we visualized the maximum genus size for every individual tree (10 000 trees per parameter set) under the three different taxonomic sorting algorithms (Supporting Information, Data S2). Under a combination of higher speciation/extinction parameters, and under higher (more lenient) threshold values, the maximum genus size does increase slightly with increasing clade size but has a clear upper threshold that is orders of magnitude lower than the clade size. Genus size is hence not saturated or constrained by simulation tree size. In simulation, the largest genus size recovered was a single instance of a genus with 675 species, under a broad threshold in IDT that was selected to examine extreme behaviour (Supporting Information, Fig. S2.2; IDT threshold = 15%). In that simulation, the frequency distribution of genus size becomes extremely flat with only 6% of species in monotypic genera, significantly diverging from patterns seen in ‘real-world’ taxonomy. The largest genera recovered under more moderate threshold values were all smaller than 350 species (Fig. 4). The distributions of genus size from RDT simulations did not change substantially with different speciation/extinction rate parameter pairs (Fig. 4A). Changes in threshold value had no substantial effect on the resulting patterns (Fig. 4A, Supporting Information, Fig. S2.2). In these simulations, two-species genera are recovered most frequently, and the second largest group is monotypic genera. This somewhat violates the expected ‘hollow curve’ where monotypic groups are otherwise the largest fraction of genera. This artefact arises from the RDT rules, in which any pair of sister species form a genus regardless of the depth of their common ancestor. However, the artefact does not appear to extend to the rest of the curve, and we note that the combined proportion of one- and two-species genera is similar across all taxonomic algorithms. While this has some implications for the use of topological criteria (discussed below), we do not consider that the overall pattern undermines the expectation of dominant monotypes in taxonomy. The proportion of monotypic genera and the size of the largest genera recovered were less sensitive to changing parameters than under either FDT or IDT. Among all the parameter sets tested, the proportion of monotypic genera ranged from 36.6 to 47.2%, and the size of the largest genera recovered ranged from 8 to 36 species per genus (Supporting Information, Figs S2.2, S2.3), closely in line with proportions in real-world taxonomy (Table 1). The IDT algorithm consistently recovered larger maximum genus sizes than the other two algorithms. Increasing rates of speciation resulted in broader and flatter genus size-frequency distributions (Fig. 4B). This ‘flattening’ decreased the left skew of the frequency distribution as evidenced in both a relatively lower proportion of monotypic species and larger maximum genus sizes. Speciation parameters at both extremes of our range of test values produce frequency distributions that deviate from the patterns seen in real-world taxonomic data. Variation in the threshold value did not alter the overall shape of the frequency distribution under any particular parameter set (Fig. 4B), but increasing the threshold value caused the same flattening effect as increasing speciation rate (Supporting Information, Fig. S2.2). The proportion of monotypic genera and the size of the largest genus covary, ranging from 6.1% monotypic with a maximum genus size of 674 species, under the highest speciation rate and highest threshold tested (λ = 0.20, threshold 15%), to up to 79.2% and a largest genus size of 10 species under the lowest parameters (λ = 0.015, threshold 3.75%). FDT recovers distribution patterns that are similar to IDT. However, FDT is much more sensitive to changes in speciation/extinction rate parameters, varying slightly more than IDT with changing speciation rates, and like IDT, an increase in speciation rate resulted in increasingly broad genus size-frequency distributions (Fig. 4C). Under all variations, the proportion of monotypic genera ranged from only 4% of genera monotypic to 79% of genera monotypic (Supporting Information, Fig. S2.2). For the lowest speciation rate applied (λ = 0.015), up to 73.7% of FDT simulated genera were monotypic under a 10% threshold, compared to 13.2% of genera monotypic under the highest speciation rate applied (λ = 0.200). FDT recovers lower maximum genus sizes than IDT. Increasing rates of speciation produced increasingly larger maximum genus sizes, ranging from ten species per genus under the lowest speciation rate to a genus with 75 species under the highest simulated speciation rate, or up to 156 species in the largest single genus from a 15% threshold (Fig. 4C). Increases in threshold values, like IDT, created the same effect on the resulting frequency distribution as increasing speciation rate parameters (Supporting Information, Fig. S2.2). Combining the data for all five speciation/extinction parameter sets provides a visualization of the central tendency of the behaviour for each algorithm (Fig. 4D). All three taxonomic algorithms produced frequency distributions that were similar to each other and strongly similar to the hollow curve distributions found in real-world taxonomy. DISCUSSION Size-frequency distributions Discussion abounds over the potential inconsistency of taxonomic delimitations (Gift & Stevens, 1997). Different organismal groups are classified with different interpretations of rank, especially comparing invertebrate and vertebrate groups (Avise & Johns, 1999; Avise & Liu, 2011). This inconsistency or apparent instability may seem to be a fundamental handicap to modernizing systematic classifications. In this context, it is interesting that the size frequency of metazoan genera converges on a strongly consistent pattern, and that pattern also agrees mathematically with distributions that emerge from idealized phylogenetic simulations. Our results demonstrate that the sizes of higher ranks behave in a predictable fashion, supporting their use as a proxy for specific diversity (taxonomic surrogacy) in synoptic studies. These patterns emerge consistently, at sufficiently large samples. Taxonomic surrogacy has many practical advantages for measuring biodiversity, which underlie the widespread use of that approach. Work on morphological disparity in living species has supported the utility of higher-ranked taxa (Triantis et al., 2016). And, even more frequently, synoptic work on the fossil record has reinforced the importance of evolutionary information from higher ranks (Raup & Boyajian, 1988). For a few well-studied groups, there is demonstrable congruence in species phylogeny and morphologically defined genera (e.g. Jablonski & Finarelli, 2009; Holt & Jønsson, 2014; Humphreys & Barraclough, 2014). These provide significant hope or reassurance that it is theoretically possible to apply traditional Linnean classifications where taxonomic ranks have a clearly articulated evolutionary or temporal delimitation. Nonetheless, the question of whether genera represent real biological or evolutionary entities has not been directly addressed outside those very few groups for which phylogenetic studies with dense taxon sampling are available. A lack of certainty about which patterns are universal or artefactual remains a persistent criticism of the transferable meaning of ranked taxonomy (Lee, 2003). The dominance of monotypic genera, and the rarity of large genera, is an established consistent pattern that has been ‘re-discovered’ repeatedly for more than a century (Aldous, 2001). Indeed, the pattern should be expected from birth–death models (Kendall, 1948). One of our taxonomic algorithms recovered a high number of two-species genera, but only under a highly unrealistic taxonomic scenario (forcing sister species to share a genus even if they deeply divergent). There is a significant body of work on the long-tailed distribution of species richness among genera (Yule, 1925; Maruvka et al., 2013), but the idea still persists that supraspecific groups are more arbitrary than species definitions and the skewed frequency distribution might be an artefact of taxonomic practice (e.g. Scotland & Sanderson, 2004; Strand & Panova, 2014). Our new data show, however, that this frequency pattern is strongly consistent across independent groups, with different taxonomic approaches and evolutionary histories. Our modelling demonstrates that it can arise from the interaction of phylogeny and taxonomy alone. The difference between taxonomic units and nested clades is a persistent misunderstanding in controversies about the utility of ranked taxonomy (Giribet et al., 2016). Even though our simulated genera are all monophyletic, the sister taxon of a genus is rarely another genus. This is not problematic; it is a reflection of the intentionally relativistic nature of ranked taxonomy. The patterns of nested clades in phylogenetic trees are informative to evolutionary processes, but they are not equivalent to taxonomy. Mathematical patterns that arise from topology have been referred to as tree ‘imbalance’ in computational phylogenetics (Mooers & Heard, 1997). Perfectly balanced bifurcating trees can only arise under very narrowly constrained circumstances, so phylogenetic imbalance, or a skew distribution in the size of daughter clades, is the expected condition and arises from random splitting in birth–death models (Nee, 2006). Metrics of tree imbalance examine nested clades; real applied taxonomy and our synthetic taxonomy are not so restricted. Even though our simulated genera are all monophyletic, the sister taxon of a genus is rarely another genus. Most phylogenetic simulations differ from patterns observed in taxonomy in that the models recover far fewer monotypic clades (Scotland & Sanderson, 2004). This is in contrast to our compiled real-world data sets, which show a consistent proportion of monotypic genera, and our simulations, which recover frequencies of monotypes that closely match real-world data (Fig. 4D). Substantial previous research has explored genus size, or more generally clade size, frequency distribution with simulation and modelling. In this context, we differentiate between what we term ‘top–down’ and ‘bottom–up’ approaches. ‘Top–down’ includes any model that directly generates the size or origination of higher taxa as units themselves. The most direct ‘top–down’ models have examined the patterns in real-world, empirical data for taxonomic classification and then derived comparable mathematical descriptions that could be used to understand underlying evolutionary patterns (e.g. Yule, 1925; Maruvka et al., 2013). Others used phylogenetic simulations from branching processes with the origination of higher taxa embedded as a term included in the model and examined the species richness of directly generated genera or ‘paraclades’ (e.g. Patzkowsky, 1995), comparing simulation results with empirical data (Przeworski & Wall, 1998; Foote, 2012). A very few prior studies used a ‘bottom–up’ approach (as we did herein), by which we mean that they first generated a simulated species phylogeny, and then applied classification. However, this approach previously was primarily used as a tool to examine cladogenesis and lineage origination over time (Sepkoski & Kendrick, 1993; Robeck, Maley & Donoghue, 2000). Our novel ‘bottom–up’ approach, or synthetic taxonomy, is the most direct approximation of the process of classifying living taxa in context of their evolutionary relationships. Previous ‘top–down’ models fitted to observed genus size distributions produced closer matches to real-world data than we obtain here through artificial taxonomy, because that was their explicit aim (Maruvka et al., 2013). Other studies have also obtained good fits to empirical data with birth–death models that include direct simulation of higher taxa as cladogenic events (Foote, 2012). By contrast, our results come from a new bottom–up approach that compares the ways that species might be partitioned into genera, given total knowledge of their phylogeny in simulation. This is an important distinction, because we are modelling the patterns of species origination, not controlling the origination of genera nor deriving a model to emulate their observed patterns. Our approach was designed to address the central question of whether human-determined, historical taxonomy can be rationalized with phylogenetic patterns. While we had no a priori expectation that synthetic phylogenetically driven taxonomy should replicate real-world data, there are clear similarities. None of the algorithms we used to recover simulated ‘genera’ were intended to closely mimic any taxonomic process. Rather we aimed to test the consistency of emergent patterns under several different idealized, monophyletic taxonomic definitions. We also used large sample sizes compared to real taxonomy, on the order of 108 simulated living species, compared to maximum global species estimates on the order of 107 (Mora et al., 2011; Scheffers et al., 2012; Stork et al., 2015). The observations and data discussed here represent large-scale emergent patterns in global biodiversity. In smaller sample sizes, the contingencies of either taxonomic history or evolutionary history could lead to the deviations that have previously been interpreted as evidence that the overall skew distributions are artefactual. Skew distributions are common in natural systems, despite great variety in underlying mechanisms for sorting objects into frequency groups. Certain standard skew distributions approximately mimic the frequencies of genera of different sizes (Reed & Hughes, 2002), as well as patterns of word frequencies in language or the sizes of corporations or cities (Reed & Jorgensen, 2004). Emergent global patterns in taxonomic diversity do not belie the many particular mechanisms that lead to the origination of large or small genera in particular clades. Large corporations are the minority of companies, but that does not mean that all large corporations are successful for the same reason(s). The same applies to the species richness of genera. Similarly, any particular explanation for the evolutionary dynamics in a particular group (a key adaptation or contraction through extinction) may not undermine its role in a larger stochastic process. Smaller samples can easily find a pattern that appears to deviate from central tendency, which has previously caused some doubt about whether this skew distribution is artefactual (e.g. Strand & Panova, 2014). We contend that the repeated finding of nearly identical patterns in taxonomic data sets at varying scales (e.g. Yule, 1925; Holman, 1985; Mora et al., 2011; Maruvka et al., 2013; Strand & Panova, 2014; and herein) is evidence that skew distribution in taxonomic size frequency is mathematically valuable. The new insight afforded by our simulations is that this is a realistic product of species evolution. The question of monophyly in real-world taxonomic data could influence patterns at multiple levels. The frequency distribution of genus size does not change when restricted to phylogenetically defined clades; we selected ‘real-world’ taxonomic data sets based on taxonomic completeness and acceptance by relevant experts, and they include both monophyletic clades (e.g. Aves) and non-monophyletic assemblages (marine invertebrates, fish). Yet the overall frequency distributions appear similar. Within each data set, most genera are defined by morphology; most genus names pre-date molecular phylogenetics, and the vast majority of species lack sequence data (Appeltans et al., 2012). Most genera and families (especially in under-studied groups) have also not been tested for monophyly, although the absence of a test does not imply that all will fail. But this pattern cannot be blamed on ‘lumping’, ‘splitting’ or cryptic species complexes. Some genera included in our data sets are undoubtedly paraphyletic, though previous simulations have shown this does not necessarily affect overall patterns, at least when including extinct lineages (Sepkoski & Kendrick, 1993). The emergence of a hollow curve distribution in real-world taxonomic data is not dependent on genera being monophyletic, yet it also emerges consistently from simulations using strict monophyly. Future generalizations about species diversity should account for the underlying frequency distribution of genus size. In a strongly skewed distribution, central tendency measures such as the arithmetic mean are relatively uninformative. Many authors (e.g. Qian & Ricklefs, 2000; Krug, Jablonski & Valentine, 2008; Mora et al., 2008; Foote, 2012) have relied on a species-per-genus ratio or used such a ratio as a proxy for maximum genus size. While many authors have discussed or made adjustments for genus size distributions, nonetheless this approach is equivalent to using an average of species per genus and implicitly assumes an underlying normal distribution for genus size. Though authors may have a thorough understanding of the taxonomic patterns within their group or even the global patterns discussed here, it should be emphasized that taxonomic metadata are applied to many other fields of science. Other work has highlighted the potential pitfalls of extrapolations based on unsubstantiated assumptions of a universal species-per-genus ratio (e.g. Scheffers et al., 2012). The modal genus size is very likely to always be one (Aldous, 2001), and the mean is hence not a useful measure of central tendency in genus diversity. Future studies can expand on the present work to estimate diversity using a modelling approach for reconstructing species diversity from a more accurate generalized probability distribution for genus diversity. Large genera Evolutionary biology is intellectually focussed on large and rapidly evolving groups (Losos et al., 1998; Thorpe & Losos, 2004; Seehausen, 2006; Rabosky & Lovette, 2008). The ‘success’ of a genus is considered nearly synonymous with its species richness (Minelli, 2015). Indeed, a substantial proportion of species are included in large genera – in reptiles, the five largest genera (Anolis, Liolaemus, Cyrtodactylus, Atractus and Hemidactylus) comprise slightly more than 10% of nominal reptile species, and the species in monotypic genera account for less than 10% of species in each of the taxonomic data sets included herein. Among relatively under-studied groups, large genera are often ‘bucket’ taxa awaiting taxonomic revision, rather than interesting evolutionary phenomena. In our data sets, there are only five genera with more than 500 species (all marine invertebrates). Some have additional structure; the gastropod Conus, for example, was recently divided into 57 subgenera (Puillandre et al., 2015). Flowering plants and fungi, not sampled here, contain some of the largest eukaryotic genera with thousands of species (Minelli, 2015); these too often have recognized additional phylogenetic structure and are split into many subgenera. Among all groups, very large genera appear to represent units that are not ‘real’ either in that they are non-monophyletic or not appropriate to the rank of genus. In order to compare like with like, across a broad range of organisms, we considered that it was better to use the taxonomic ranks assigned by experts rather than imposing our own re-interpretation. For instance, some groups have subgeneric divisions that could arguably be the equivalent to the genera of other groups; we did not impose this equivalence as it would involve overturning the decision of experts as to what relevant level of distinctiveness is required to differentiate a genus in that group. It is interesting then that using a sampling of the current taxonomic status quo recovered consistent patterns of genus size distribution across all the animal groups we investigated. The main goal of our study was to determine whether taxonomic rank in general, but genera in particular, can predict species biodiversity; one immediate outcome is that our findings can be used to assess where biological groups may deviate from that null model. We suggest, for example, that this is a further evidence to support critical re-examination of unusually large genera especially among marine invertebrates, and unusually high frequencies of small genera, such as in mammals. Rates of evolution There are real, predictable patterns in systematics, and the skew distribution in generic size occurs across variety of rate parameters and taxonomic algorithms. Our simulations deliberately used fixed rates of speciation and extinction to facilitate comparisons between rate parameters; this led to a well-constrained behaviour in the resulting trees. There is a clear mathematical behaviour to trees, influenced by speciation and extinction rates, which translates to mathematical behaviours of clades (Aldous et al., 2008). Our taxonomic algorithms were also deliberately defined in an idealized way that is not realistically similar to practical taxonomy. Taxonomy almost always operates with limited data, inferring relationships based on key characters with established utility (whether molecular or morphological), as available for the specimens under study. In simulation, we have omniscient knowledge of the underlying phylogeny, so this provides a way to assess how constrained or variable genus size frequency would be, in comparing perfectly complete and accurate phylogenies under a range of evolutionary rates. Our first approach to simulated taxonomy, RDT (Fig. 4B), extends the phylogenetic species concept so that ranks are assigned based on the relative similarity of proximate monophyletic groupings (sensuCracraft, 1983). The second approach, IDT (Fig. 4C), is conceptually similar in that it separates clusters of taxa where they have diverged ancestrally for more than some fixed threshold of time. FDT (Fig. 4A) approximates the chronological approaches promoted by some authors, who advocate the use of divergence times to determine rank (Avise & Johns, 1999; Avise & Mitchell, 2007). It should be expected that FDT simulations would deviate from ‘real-world’ taxonomy because this is not how taxa are defined in practice; however, it may be successfully applied post hoc to a well-resolved phylogeny (Holt & Jønsson, 2014). Lineage depth is of interest in delimiting taxonomic groups, but it is not information that is generally accessible or available for most species-level taxa (Ricotta et al., 2012). Age of origin is variable in different groups – a topological phenomenon that is explored in our other taxonomic algorithms – and information that is simply not known for many. This potential problem has been well known for decades (e.g. Hennig, 1979; Avise & Liu, 2011). Our simulations demonstrate that the FDT approach is highly sensitive to permutations of speciation and extinction rates (Fig. 4C), whereas ‘real-world’ taxonomy is evidentially not, at comparable sampling magnitudes. Small changes in evolutionary rates caused the FDT and IDT simulations to shift away from biologically realistic distributions. More importantly perhaps, different depth (age) thresholds actually had relatively less impact on the resulting frequency distributions. This sensitivity illustrates a significant weakness in using time of origin as a criterion for defining higher taxa. Under the RDT model, varying rate parameters had very limited impact on frequency distributions, even less variable than in the real-world data. While RDT is also not intended to mimic genuine taxonomic practice, this pattern demonstrates that similarly shaped distributions can arise directly from different evolutionary scenarios, which is undoubtedly the case in comparing groups of real organisms. This method still uses branch lengths as well as topology to define genera (Barraclough & Humphreys, 2015), yet recovers rather different frequency distributions. The large number of bitypic genera recovered by RDT is an artefact reflecting the effects of forcing the classification to seek sister-relationships even when those taxa may be separated by deep divergences. In real species, characterized by genetic or morphological characters, deeply separated sister taxa would probably not be considered a bitypic genus but rather two monotypic genera. The three taxonomic algorithms we used to classify our simulated trees usually recovered genera that had smaller maximum sizes than in ‘real-world’ data. Large genera in some cases reflect the existence of ‘bucket’ para- or polyphyletic genera in real-world taxonomy; these are never present in our simulations. Other very large genera in the real world are undoubtedly monophyletic and may be already subdivided into subgenera, which may in fact be more equivalent to the genus rank in other clades (e.g. the mollusc genus Conus, noted above). More likely, large genera may be absent in the simulations because the model did not allow for synergistic effects of speciation rates and environment, which are thought to underpin rapid radiations (Harmon & Harrison, 2015). It is increasingly well understood that both speciation rates and extinction rates vary among clades and even within clades over time (Marshall, 2017), although these rates may be approximately equal (zero net diversification) across all clades over time (Ricklefs, 2007) or with a narrow tendency for globally increasing diversity (Bennett, 2013). The convergence of genus size-frequency distributions under our various models and the similar convergence in real-world taxonomic data suggest that there is perhaps a long-term equilibrium in evolutionary rates. Recent work has highlighted the potential heritability of speciation as a trait itself (e.g. Purvis et al., 2011; Rabosky & Goldberg, 2015). The constrained sizes of the largest genera recovered from our simulations with fixed speciation rates provide strong additional evidence that heterogeneous rates of speciation are fundamental to the origination of large genera. There are two significant hurdles that have been raised as potentially impeding the use of higher-ranked taxa to measure species diversity: First, whether the units (genera) are defined by consistent criteria that make them comparable across different groups, and second, whether the genera are monophyletic units (Hendricks et al., 2014). Our simulations addressed these issues by using strict algorithms to define monophyletic genera. Applying these criteria highlighted the variability introduced by changing evolutionary rates and also illustrated the comparatively constrained range of distributions found in real-world taxonomy. CONCLUSIONS Mathematical approaches are important tools to separate real excursions in speciation rates, that might require special explanations, from patterns that can be predicted within a well-described probability distribution. If we begin with a premise that large genera represent evolutionary anomalies, then it is logical to seek an explanation for the process that generated that excursion. However, as we demonstrate here, taxonomic genera arise from phylogeny in a probability space that accommodates both small and large genera, with decreasing frequency as genera get larger. From these simulations, one could infer that genera of sizes up to around 50 species are not exceptional, genera of several hundred species are unusual and perhaps deserve taxonomic scrutiny, and certainly monotypic genera are commonplace. Special adaptive significance is not necessarily required to explain a monotypic genus, or a large genus, or a genus with four species. Our results provide novel evidence that Linnean ranks applied to groups of species can have transferable meaning between unrelated clades, even though monotypic units of classification are not equivalent to topological nested clades. Genus sizes should follow a skew distribution; monotypic genera are expected to be very common, and large genera are expected to be very rare. The largest genera, of sizes that dramatically exceed anything recovered in simulation, are probably not appropriate phylogenetic or systematic units. Understanding the frequency distribution of supraspecific taxa, and their behaviour as mathematical units, is crucial to a more robust understanding of taxonomic surrogacy. It is essential to know how diversity, when measured in terms of genera or families, can be translated into species richness. The skewed distribution of genus sizes, which is a real phenomenon, precludes using a simple count of genera or higher-ranked taxa to answer many questions about comparative species diversity. The present study provides a foundation for a new approach to quantify the error introduced by taxonomic surrogacy. Our results demonstrate for the first time that determining this is an achievable target, and that established systematics already holds the key to robust quantitative analyses of global diversity. SUPPORTING INFORMATION Additional Supporting Information may be found in the online version of this article at the publisher’s website: Data S1. ‘MBL 2017’: Software used to generate simulated phylogenetic trees and synthetic taxonomy. The package contains 15 files. MBL2017 can be executed on PC or Mac but requires the Qt library (www.qt.io). Data S2. Supplementary explanation of results, including description of taxonomic sorting algorithms, example taxonomically sorted output from tree simulations, data quality approach to real-world taxonomic data, frequency distributions from simulated data and ‘real-world’ data, and quantitative comparisons among real-world data sets. ACKNOWLEDGEMENTS This research was supported by the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement no. H2020-MSCA-IF-2014-655661 to JDS). Amy Garbett and Bernard Picton (Queen’s University Marine Laboratory) assisted with an earlier version of this study. 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