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

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 $1-o(1)$.}