Enumeration of coalescent histories for caterpillar species trees and pp-pseudocaterpillar gene trees

Abstract

For a fixed set XX containing nn taxon labels, an ordered pair consisting of a gene tree topology GG and a species tree SS bijectively labeled with the labels of XX possesses a set of coalescent histories -- mappings from the set of internal nodes of GG to the set of edges of SS describing possible lists of edges in SS on which the coalescences in GG take place. Enumerations of coalescent histories for gene trees and species trees have produced suggestive results regarding the pairs (G,S)(G,S) that, for a fixed nn, have the largest number of coalescent histories. We define a class of 2-cherry binary tree topologies that we term pp-pseudocaterpillars, examining coalescent histories for non-matching pairs (G,S)(G,S), in the case in which SS has a caterpillar shape and GG has a pp-pseudocaterpillar shape. Using a construction that associates coalescent histories for (G,S)(G,S) with a class of "roadblocked" monotonic paths, we identify the pp-pseudocaterpillar labeled gene tree topology that, for a fixed caterpillar labeled species tree topology, gives rise to the largest number of coalescent histories. The shape that maximizes the number of coalescent histories places the "second" cherry of the pp-pseudocaterpillar equidistantly from the root of the "first" cherry and from the tree root. A symmetry in the numbers of coalescent histories for pp-pseudocaterpillar gene trees and caterpillar species trees is seen to exist around the maximizing value of the parameter pp. The results provide insight into the factors that influence the number of coalescent histories possible for a given gene tree and species tree

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