Universal height and width bounds for random trees

TL;DR

This paper establishes universal tail bounds for the height and width of various random trees, proving conjectures and solving open problems by analyzing degree distributions and using probabilistic techniques.

Contribution

It introduces a novel approach linking degree paths to size-biased degree orderings, and adapts Poissonization methods to derive bounds for random trees with fixed degrees.

Findings

Proved non-asymptotic tail bounds for tree heights.

Resolved conjectures and open problems in the field.

Unified bounds applicable to multiple random tree models.

Abstract

We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degree statistics, and a random truncation of a size-biased ordering of the degrees of such a tree. We also exploit a Poissonization trick introduced by Camarri and Pitman (https://doi.org/10.1214/EJP.v5-58) in the context of inhomogeneous continuum random trees, which we adapt to the setting of random trees with fixed degrees. Finally, we propose and justify a change to the conventions of branching process nomenclature: the name "Galton-Watson trees" should be permanently retired by the community, and replaced with the name "Bienaym\'e trees".