Repeated fringe subtrees in random rooted trees

Abstract

A fringe subtree of a rooted tree is a subtree that consists of a node and all its descendants. In this paper, we are particularly interested in the number of fringe subtrees that occur repeatedly in a random rooted tree. Specifically, we show that the average number of fringe subtrees that occur at least r times is of asymptotic order n/(log n)3/2 for every r ≥ 2 (with small periodic fluctuations in the main term) if a tree is taken uniformly at random from a simply generated family. Moreover, we also prove a strong concentration result for a related parameter: the size of the smallest tree that does not occur as a fringe subtree is with high probability equal to one of at most two different values. The main proof ingredients are singularity analysis, bootstrapping and the first and second moment methods.