Limit distribution for the maximum degree of a random recursive tree

If a recursive tree is selected uniformly at random from among all recursive trees on n vertices, then the distribution of the maximum in-degree is given asymptotically by the following theorem: for any 3xed integer d, Pn( 6 n + d)= exp(−2{ n}−d−1) + o(1) as n → ∞; where n = log2 n. (As usual, n denotes the greatest integer less than or equal to n, and { n}= n− n :) The proof makes extensive use of asymptotic approximations for the partial sums of the exponential series. c © 2002 Elsevier Science B.V. All rights reserved.