Coalescent Random Forests

Abstract

Various enumerations of labeled trees and forests, including Cayley's formulann-2for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the followingcoalescent constructionof a sequence of random forests (Rn,Rn-1,...,R1) such thatRkhas uniform distribution over the set of all forests ofkrooted trees labeled by [n]. LetRnbe the trivial forest withnroot vertices and no edges. Forn≥k≥2, given thatRn,...,Rkhave been defined so thatRkis a rooted forest ofktrees, defineRk-1by addition toRkof a single edge picked uniformly at random from the set ofn(k-1) edges which when added toRkyield a rooted forest ofk-1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, theadditive coalescentin which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitudesxandyrunning a risk at ratex+yof a coalescent collision resulting in a mass of magnitudex+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.