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 ≥ 0 is shown to decay exponentially with k. Notoriously hard to compute, the exact fractions c k 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 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.