On a random search tree: asymptotic enumeration of vertices by distance from leaves

A random binary search tree grown from the uniformly random permutation of $[n]$ is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction $c_k$ of vertices of a fixed rank $k\ge 0$ is shown to decay exponentially with $k$. Notoriously hard to compute, the exact fractions $c_k$ had been determined for $k\le 3$ only. We computed $c_4$ and $c_5$ as well; both are ratios of enormous integers, denominator of $c_5$ being $274$ digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor of $c_k$’s denominator is $2^{k+1}+1$ at most. We conjecture that, in fact, the prime divisors of every denominator for $k>1$ form a single interval, from $2$ to the largest prime not exceeding $2^{k+1}+1$.