Random trees have height $O(\sqrt{n})$

TL;DR

This paper establishes new tail bounds for the height of various classes of random trees, confirms three conjectures, and introduces a stochastic ordering on tree heights based on degree sequences.

Contribution

It provides the first non-asymptotic tail bounds for the height of random trees with given degree sequences and introduces a stochastic ordering framework.

Findings

Tail bounds for heights of various random trees

Confirmation of three conjectures by Janson (2012)

Stochastic ordering of tree heights based on degree sequences

Abstract

We obtain new non-asymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaym\'e trees (the family trees of branching processes), in the process settling three conjectures of Janson (2012) and answering several other questions from the literature. Moreover, we define a partial ordering on degree sequences and show that it induces a stochastic ordering on the heights of uniformly random trees with given degree sequences. The latter result can also be used to show that sub-binary random trees are stochastically the tallest trees with a given number of vertices and leaves (and thus that random binary trees are the stochastically tallest random homeomorphically irreducible trees with a given number of vertices). Our proofs are based in part on the bijection between trees and sequences introduced by Foata and Fuchs (1970), which can be recast to provide a line-breaking construction of random trees with given vertex degrees as shown in Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin (2023).