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 ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. Notoriously hard to compute, the exact fractions ck had been determined for k ≤ 3 only. We computed c4 and c5 as well; both are ratios of enormous integers, denominator of c5 being 274 digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor of ck’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 + 1.