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

• Authors: L. Addario-Berry
• Venue: Random Structures & Algorithms
• Year: 2012

Fix a sequence c = (c1,…,cn) 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 v1,…,vn so that for each 1 ≤ i ≤ n, vi has exactly ci children. Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document} **image** 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 \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document} **image**. These bounds are optimal up to the constant in the exponent when c satisfies \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}$\sum_{i=1}^n c_i^2=O(n)$\end{document} **image**; the latter can be viewed as a “finite variance” condition for the child sequence. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.