The height of scaled attachment random recursive trees

We study depth properties of a general class of random recursive trees where each node n attaches to the random node !nXn" and X0, . . . , Xn is a sequence of i.i.d. random variables taking values in [0, 1). We call such trees scaled attachment random recursive trees (SARRT). We prove that the height Hn of a SARRT is asymptotically given by Hn ∼ αmax log n where αmax is a constant depending only on the distribution of X0 whenever X0 has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees Hn ∼ e log n that does not use branching random walks.