Coalescent Random Forests (1998)
| Venue: | J. COMBINATORIAL THEORY A |
| Citations: | 53 - 15 self |
BibTeX
@MISC{Pitman98coalescentrandom,
author = {Jim Pitman},
title = {Coalescent Random Forests},
year = {1998}
}
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Abstract
Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 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 (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 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 magnitude...
Keyphrases
coalescent random forest rooted forest multinomial expansion following coalescent construction physical process binary coalescent collision single edge root vertex uniform distribution additive coalescent define gamma1 labeled tree trivial forest various enumeration coalescent construction formula gamma2 random forest
