Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees

TL;DR

This paper establishes near-optimal sub-Gaussian tail bounds for the height, width, and level-wise node counts of conditioned Galton--Watson trees with critical offspring distribution, enhancing understanding of their probabilistic structure.

Contribution

It provides the first sub-Gaussian tail bounds for height and width of conditioned Galton--Watson trees, with bounds that are optimal up to constants.

Findings

Sub-Gaussian tail bounds for height and width are derived.

Bounds are optimal up to constant factors.

Upper tail bounds for nodes at level k are established.

Abstract

We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.