Coalescent Random Forests

Various enumerations of labeled trees and forests, including Cayleyʹs formulann−2for the number of trees labeled by [n], and Cayleyʹs 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.