Regular Article
Coalescent Random Forests,☆☆

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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. Fornk⩾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.

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Research supported in part by N.S.F. Grants MCS94-04345 and DMS 97-03961.

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