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

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.