Asymptotic enumeration of compacted binary trees of bounded right height

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:172

Article

Genitrini, Antoine¹  Gittenberger, Bernhard²  Kauers, Manuel³  Wallner, Michael² 

[1] Sorbonne Univ, LIP6, CNRS, UMR 7606, F-75005 Paris, France

[2] Tech Univ Wien, Wiedner Hauptstr 8-10-104, A-1040 Vienna, Austria

[3] Johannes Kepler Univ Linz, Inst Algebra, Altenberger Str 69, A-4040 Linz, Austria

关键词: Compacted trees;    Enumeration;    D-finiteness;    Analytic combinatorics;    Directed acyclic graphs;    Chebyshev polynomials;   

DOI  :  10.1016/j.jcta.2019.105177

来源: Elsevier

【 摘 要 】

A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf. We solve the asymptotic counting problem for this class as well as a closely related, further simplified class. For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations. Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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