Critical trees are neither too short nor too fat

TL;DR

This paper derives optimal probabilistic bounds for the height and width of critical Bienaymé trees conditioned on size, and explores asymptotics for certain offspring distributions, advancing understanding of their geometric properties.

Contribution

It provides the first optimal tail bounds for height and width of critical size-conditioned Bienaymé trees and analyzes asymptotics for distributions in the Cauchy domain of attraction.

Findings

Optimal lower tail bounds for height

Optimal upper tail bounds for width

Asymptotic behavior for Cauchy domain offspring distributions

Abstract

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym\'e trees.