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We prove that for any fixed $k$, the probability that a random vertex of arandom increasing plane tree is of rank $k$, that is, the probability that arandom 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 thelimiting rank distribution is super-exponentially narrow. We prove that thelatter property holds uniformly for all finite $n$ as well.} More generally, weprove that the ranks of a finite uniformly random set of vertices areasymptotically independent, each with distribution $\{c_k\}$. We compute theexact value of $c_k$ for $0\leq k\leq 3$, demonstrating that the limitingexpected fraction of vertices with rank $\le 3$ is $0.9997\dots$. We show thatwith probability $1-n^{-0.99\eps}$ the highest rank of a vertex in the tree issandwiched between $(1-\eps)\log n /\log\log n$ and $(1.5+\eps)\log n/\log\logn$, {\color{blue} and that this rank is asymptotic to $\log n/\log\log n$ withprobability $1-o(1)$.}

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