• random trees;
  • height;
  • power of choice;
  • renewal process;
  • second moment method


We study depth properties of a general class of random recursive trees where each node i attaches to the random node equation image 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 typical depth Dn, the maximum depth (or height) Hn and the minimum depth Mn of a sarrt are asymptotically given by Dn ∼μ-1 log n, Hn ∼ αmax log n and Mn ∼ αmin log n where μ,αmax and αmin are constants depending only on the distribution of X0 whenever X0 has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees Hnelog n that does not use branching random walks.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011