Discrete Mathematics & Theoretical Computer Science, Vol 7 (2005)

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DMTCS vol 7 no 1 (2005), pp. 313-400


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
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
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
corresponds to the number of binary (unlabeled and ordered) trees,
, of given path length
(and also the number of distinct Lempel-Ziv'78 parsings of length
sequences). We first show that the number of binary trees with given path length
is asymptotically equal to
~ 2
. 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
. 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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