Tail bounds for the height and width of a random tree with a given degree sequence

TL;DR

This paper establishes sub-Gaussian tail bounds for the height and width of a uniformly random plane tree with a specified child sequence, under a finite variance condition, providing near-optimal probabilistic bounds.

Contribution

It introduces sub-Gaussian tail bounds for tree height and width based on the child's degree sequence, extending understanding of random tree structures.

Findings

Tail bounds are optimal up to constants when degree sequence variance is finite.

Provides probabilistic bounds for height and width of random trees.

Applicable to trees with degree sequences satisfying a finite variance condition.

Abstract

Fix a sequence c=(c_1,...,c_n) of non-negative integers with sum n-1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v_1,...,v_n so that for each 1 <= i <= n, v_i has exactly c_i children. Let T be a plane tree drawn uniformly at random from among all plane trees with child sequence c. In this note we prove sub-Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of T. These bounds are optimal up to the constant in the exponent when c satisfies c_1^2+...+c_n^2=O(n); the latter can be viewed as a "finite variance" condition for the child sequence.