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

Abstract

We prove that for any fixed k, the probability that a random vertex of a random increasing plane tree is of rank k, that is, the probability that a random vertex is at distance k from the leaves, converges to a constant c k as the size n of the tree goes to infinity. We prove that 1− ∑ j≤k c k < 2 2k+3 (2k+4)! , so that the tail of the limiting rank distribution is super-exponentially narrow. We prove that the latter property holds uniformly for all finite n as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution {c k }. We compute the exact value of c k for 0 ≤ k ≤ 3, demonstrating that the limiting expected fraction of vertices with rank ≤ 3 is 0.9997… We show that with probability 1 − n −0.99𝜀 the highest rank of a vertex in the tree is sandwiched between (1 − 𝜀) log n∕ log log n and (1.5 + 𝜀) log n∕ log log n, and that this rank is asymptotic to log n∕ log log n with probability 1 − o(1).