Coalescent Random Forests

Various enumerations of labeled trees and forests, including Cayley's formula n for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (Rn , Rn&1 , ..., R1) such that Rk has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let Rn be the trivial forest with n root vertices and no edges. For n k 2, given that Rn , ..., Rk have been defined so that Rk is a rooted forest of k trees, define Rk&1 by addition to Rk of a single edge picked uniformly at random from the set of n(k&1) edges which when added to Rk yield a rooted forest of k&1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitudes x and y running a risk at rate x+ y of a coalescent collision resulting in a mass of magnitude x+ y. The transition semigroup of the additive coalescent is shown to involve probability distributions associated with a multinomial expansion over rooted forests. 1999 Academic Press