Depth Properties of scaled attachment random recursive trees

Abstract

AbstractWe study depth properties of a general class of random recursive trees where each node i attaches to the random node \documentclass{article} \usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty} \begin{document} \begin{align*}\left\lfloor iX_i\right\rfloor\end{align*} \end{document} 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 Hn ∼ elog n that does not use branching random walks.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011