The distribution of height and diameter in random non-plane binary trees

TL;DR

This paper provides a detailed probabilistic analysis of the height and diameter distributions of random non-plane binary trees, revealing their limiting behaviors and deviation estimates through complex analysis of generating functions.

Contribution

It introduces precise distributional results for the height and diameter of Otter trees, including their limiting theta distributions and deviation estimates, using complex analysis techniques.

Findings

Height follows a limiting theta distribution

Diameter distribution derived from height analysis

Provides deviation estimates for tree height and diameter

Abstract

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height.