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

TL;DR

This paper analyzes the distribution of vertex distances from leaves in random increasing plane trees, showing that these distances converge to a limiting distribution with super-exponentially narrow tail and deriving asymptotic properties of maximum ranks.

Contribution

It provides the first asymptotic enumeration of vertex ranks in increasing plane trees, including explicit calculations and probabilistic bounds for maximum vertex rank.

Findings

Probability of rank k converges to a constant c_k as tree size grows.

Tail of the rank distribution is super-exponentially narrow.

Maximum vertex rank scales as log n / log log n with high probability.

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. {\color{blue} We prove that $1-\sum_{j\le k} c_k<\tfrac{3^{k+1}}{(2k+1)!}$, 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 $0\leq k\leq 3$, demonstrating that the limiting expected fraction of vertices with rank $\le 3$ is $0.9997\dots$. We show that with probability $1-n^{-0.99\eps}$ the highest rank of a vertex in the tree is sandwiched between $(1-\eps)\log n /\log\log n$ and $(1.5+\eps)\log n/\log\log n$, {\color{blue} and that this rank is asymptotic to $\log n/\log\log n$ with probability $1-o(1)$.}