Combinatorics and probabilities of Kingman coalescent trees: histories, shapes, topologies, root splits, Colless, and Sackin
This entry formalizes combinatorics and probabilities of trees under Kingman's coalescent, through its embedded jump chain, in which each pair of the k current lineages is the next to merge with probability 1/binomial(k, 2). A ranked labelled history on n leaves is a sequence of n − 1 merges starting from n singletons; there are n!(n − 1)!/2^(n−1) of them, and each has probability 2^(n−1)/(n!(n − 1)!). Ranked and unranked tree shapes are defined as classes of histories under relabelling of the leaves; their numbers satisfy André's recurrence for alternating permutations, shifted by one, and…
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