Summary: 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 \(k\geq 0\) is shown to decay exponentially with \(k\). We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution \(\{c_k\}\). Notoriously hard to compute, the exact fractions \(c_k\) have been determined for \(k\leq 3\) only. We present a shortcut enabling us to compute \(c_4\) and \(c_5\) as well; both are ratios of enormous integers, the denominator of \(c_5\) being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of \(c_k\) is at most \(2^{k+1} + 1\). 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 \(2^{k+1} + 1\).