On the peel number and the leaf-height of Galton-Watson trees

Luc Devroye, Marcel K. Goh, Rosie Y. Zhao

Abstract: We study several parameters of a random Bienaymé–Galton–Watson tree of size defined in terms of an offspring distribution with mean and nonzero finite variance . Let be the generating function of the random variable . We show that the independence number is in probability asymptotic to , where is the unique solution to . One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to . Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If , then we show that the maximum leaf-height over all nodes in is in probability asymptotic to . If and is the first integer with , then the leaf-height is in probability asymptotic to .