The height of random binary unlabelled trees

TL;DR

This paper analyzes the height distribution of random unlabelled binary trees, showing it converges to a theta distribution and providing deviation estimates, using complex analysis of generating functions.

Contribution

It introduces a detailed analysis of the height distribution of unlabelled binary trees, including limiting distribution and deviation bounds, which was not previously established.

Findings

Height converges to a theta distribution

Derived moderate and large deviation estimates

Used complex analysis of generating functions

Abstract

This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height.