Aequat. Math. 92 (2018), 311–353
The Author(s) 2018. This article is an
open access publication
published online January 23, 2018
Fringe analysis of plane trees related to cutting and pruning
, Clemens Heuberger
, and Helmut Prodinger
Abstract. Rooted plane trees are reduced by four diﬀerent operations on the fringe. The
number of surviving nodes after reducing the tree repeatedly for a ﬁxed number of times
is asymptotically analyzed. The four diﬀerent operations include cutting all or only the
leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results
include exact expressions and asymptotic expansions for the expected value and the variance
as well as central limit theorems.
Mathematics Subject Classiﬁcation. 05A16, 05C05, 05A15, 05A19, 60C05.
Keywords. Plane trees, Pruning, Tree reductions, Central limit theorem, Narayana polyno-
Plane trees are among the most interesting elementary combinatorial objects;
they appear in the literature under many diﬀerent names such as ordered trees,
planar trees, planted plane trees, etc. They have been analyzed from various
aspects, especially due to their relevance in Computer Science. Two particu-
larly well-known quantities are the height, since it is equivalent to the stack size
needed to explore binary (expression) trees, and the pruning number (pruning
index), since it is equivalent to the register function (Horton–Strahler number)
of binary trees. Several results for the height of plane trees can be found in
Benjamin Hackl and Clemens Heuberger are supported by the Austrian Science Fund
(FWF): P 24644-N26 and by the Karl Popper Kolleg “Modeling-Simulation-Optimization”
funded by the Alpen-Adria-Universit¨at Klagenfurt and by the Carinthian Economic Pro-
motion Fund (KWF). Helmut Prodinger is supported by an incentive grant of the National
Research Foundation of South Africa. Part of this author’s work was done while he visited
Academia Sinica. He thanks the Institute of Statistical Science for its hospitality.
This is the full version of the extended abstract .