Congruence properties of depths in some random trees

Consider a random recursive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n → ∞. The same is true for the number of vertices of depth divisible by m for m = 3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m ≥ 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m = 6 is intermediate: the number is asymptotically normal but the variance is of order n log n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and explain, this non-intuitive behaviour using a translation to a generalized Pólya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m ≤ 8 but not for m ≥ 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton–Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.