Depth Properties of scaled attachment random recursive trees

We study depth properties of a general class of random recursive trees where each node i attaches to the random node biXic 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 logn where αmax is a constant depending only on the distribution of X0 whenever X0 has a density. This gives a new elementary proof for the height of uniform random recursive trees Hn ∼ e logn that does not use branching random walks.