Simply Generated Trees, Conditioned Galton–watson Trees, Random Allocations and Condensation: Extended Abstract

taking the product over all nodes v in T , where d+(v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let Tn be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w0 > 0 and that there exists some k > 2 with wk > 0. We consider only n such that there exists some tree with n vertices and positive weight.) One particularly important case is when ∑∞ k=0wk = 1, so the weight sequence (wk) is a probability distribution on Z>0. (We then say that (wk) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = wk. It is easily seen that the simply generated random tree Tn equals the conditioned Galton–Watson tree with offspring distribution ξ, i.e., the random Galton–Watson tree defined by ξ conditioned on having exactly n vertices. One of the reasons for the interest in these trees is that many kinds of random trees occuring in various applications (random ordered trees, unordered trees, binary trees, . . . ) can be seen as simply generated random trees and conditioned Galton–Watson trees, see e.g. Aldous [3, 4], Devroye [9] and Drmota [10]. It is easily seen that if a, b > 0 and we change wk to w̃k := ab wk, (1.2)