The Unconstrained Diameters of the Duplication-Loss Cost and the Loss Cost

Tree reconciliation costs are a popular choice to account for the discordance between the evolutionary history of a gene family (i.e., a gene tree), and the species tree through which this family has evolved. This discordance is accounted for by the minimum number of postulated evolutionary events necessary for reconciling the two trees. Such events include gene duplication, loss, and deep coalescence, and are used to define different types of tree reconciliation costs. For example, the duplication-loss cost for a gene tree and species tree accounts for the minimum number of gene duplications and losses necessary to reconcile these trees. Fundamental to the understanding of how gene trees and species trees relate to each other are the diameters of tree reconciliation costs. While such diameters have been well-researched, still absent from these studies are the unconstrained diameters for two of the classic tree reconciliation costs, namely the duplication-loss cost and the loss cost. Here, we show the essential mathematical properties of these diameters and provide efficient solutions for computing them. Finally, we analyze the distributions of these diameters using simulated datasets.