Tight universal bounds on the height times the width of random trees

TL;DR

This paper establishes tight, assumption-free bounds on the product of height and width of various classes of random trees, showing it is typically on the order of n log n, thus answering a previously open question.

Contribution

It provides the first non-asymptotic, uniform bounds on the height-width product for multiple types of random trees, generalizing and tightening previous results.

Findings

The height-width product is O(n log n) with high probability.

The bound is tight in the general setting considered.

Answers a question posed by Addario-Berry (2019).

Abstract

We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaym\'e trees and simply generated trees. We show that for a tree of size $n$, this product is $O(n \log n)$ in probability, answering a question by Addario-Berry (2019). The order of this bound is tight in this generality.