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

We prove that for any fixed k$$ k $$ , the probability that a random vertex of a random increasing plane tree is of rank k$$ k $$ , that is, the probability that a random vertex is at distance k$$ k $$ from the leaves, converges to a constant ck$$ {c}_k $$ as the size n$$ n $$ of the tree goes to infinity. We prove that 1−∑j≤kck<22k+3(2k+4)!$$ 1-{\sum}_{j\le k}{c}_k<\frac{2^{2k+3}}{\left(2k+4\right)!} $$ , 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$$ n $$ as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution {ck}$$ \left\{{c}_k\right\} $$ . We compute the exact value of ck$$ {c}_k $$ for 0≤k≤3$$ 0\le k\le 3 $$ , demonstrating that the limiting expected fraction of vertices with rank ≤3$$ \le 3 $$ is 0.9997$$ 0.9997 $$ … We show that with probability 1−n−0.99ε$$ 1-{n}^{-0.99\varepsilon } $$ the highest rank of a vertex in the tree is sandwiched between (1−ε)logn/loglogn$$ \left(1-\varepsilon \right)\log n/\mathrm{loglog}n $$ and (1.5+ε)logn/loglogn$$ \left(1.5+\varepsilon \right)\log n/\mathrm{loglog}n $$ , and that this rank is asymptotic to logn/loglogn$$ \log n/\mathrm{loglog}n $$ with probability 1−o(1)$$ 1-o(1) $$ .