Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

Abstract

AbstractWe prove that for any fixed , the probability that a random vertex of a random increasing plane tree is of rank , that is, the probability that a random vertex is at distance from the leaves, converges to a constant as the size of the tree goes to infinity. We prove that , so that the tail of the limiting rank distribution is super‐exponentially narrow. We prove that the latter property holds uniformly for all finite as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution . We compute the exact value of for , demonstrating that the limiting expected fraction of vertices with rank is … We show that with probability the highest rank of a vertex in the tree is sandwiched between and , and that this rank is asymptotic to with probability .