Coalescent histories for lodgepole species trees
Filippo Disanto, Noah A. Rosenberg

TL;DR
This paper introduces lodgepole species trees and precisely counts their coalescent histories, revealing super-exponential growth and improving bounds relevant for phylogenetic calculations.
Contribution
It provides the first exact enumeration of coalescent histories for lodgepole species trees and demonstrates their super-exponential growth rate.
Findings
Number of coalescent histories grows as m!! for lodgepole trees.
Established a new lower bound for the ratio of maximum to minimum coalescent histories.
Shows growth in coalescent histories can be faster than exponential.
Abstract
Coalescent histories are combinatorial structures that describe for a given gene tree and species tree the possible lists of branches of the species tree on which the gene tree coalescences take place. Properties of the number of coalescent histories for gene trees and species trees affect a variety of probabilistic calculations in mathematical phylogenetics. Exact and asymptotic evaluations of the number of coalescent histories, however, are known only in a limited number of cases. Here we introduce a particular family of species trees, the \emph{lodgepole} species trees , in which tree has taxa. We determine the number of coalescent histories for the lodgepole species trees, in the case that the gene tree matches the species tree, showing that this number grows with in the number of taxa . This computation demonstrates the existence…
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