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The Distribution of Heights of Binary Trees and Other Simple Trees

Published online by Cambridge University Press:  12 September 2008

Philippe Flajolet ,
Zhicheng Gao ,
Andrew Odlyzko  and
Bruce Richmond
Philippe Flajolet
Affiliation:
INRIA, 78150 Rocquencourt, France
Zhicheng Gao
Affiliation:
Dept. of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S5B6, Canada
Andrew Odlyzko
Affiliation:
AT& T Bell Laboratories, Murray Hill, New Jersey 07974, United States
Bruce Richmond
Affiliation:
Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada
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Abstract

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfy

uniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 2 , June 1993 , pp. 145 - 156
Copyright
Copyright © Cambridge University Press 1993

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