DMTCS

Volume 7

n° 1 (2005), pp. 313-400

author:Charles Knessl and Wojciech Szpankowski
title:Enumeration of Binary Trees and Universal Types
keywords:Binary trees, types, Lempel-Ziv'78, path length
abstract:Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with
n
nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a given path length (sum of depths) are there? This question arose in the study of universal types of sequences. Two sequences of length
p
have the same universal type if they generate the same set of phrases in the incremental parsing of the Lempel-Ziv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length
p
corresponds to the number of binary (unlabeled and ordered) trees,
T
p
, of given path length
p
(and also the number of distinct Lempel-Ziv'78 parsings of length
p
sequences). We first show that the number of binary trees with given path length
p
is asymptotically equal to
T
p
~ 2
2p/(log
2
p)(1+O(log
-2/3
p))
. Then we establish various limiting distributions for the number of nodes (number of phrases in the Lempel-Ziv'78 scheme) when a tree is selected randomly among all trees of given path length
p
. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics.
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reference: Charles Knessl and Wojciech Szpankowski (2005), Enumeration of Binary Trees and Universal Types, Discrete Mathematics and Theoretical Computer Science 7, pp. 313-400
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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