2 research outputs found

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

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    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

    A compendium of covariances and correlation coefficients of coalescent tree properties

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    Gene genealogies are frequently studied by measuring properties such as their height (HH), length (LL), sum of external branches (EE), sum of internal branches (II), and mean of their two basal branches (BB), and the coalescence times that contribute to the other genealogical features (TT). These tree properties and their relationships can provide insight into the effects of population-genetic processes on genealogies and genetic sequences. Here, under the coalescent model, we study the 15 correlations among pairs of features of genealogical trees: HnH_n, LnL_n, EnE_n, InI_n, BnB_n, and TkT_k for a sample of size nn, with 2≀k≀n2 \leq k \leq n. We report high correlations among HnH_n, LnL_n, In,I_n, and BnB_n, with all pairwise correlations of these quantities having values greater than or equal to 6[6ΞΆ(3)+6βˆ’Ο€2]/(Ο€18+9Ο€2βˆ’Ο€4)β‰ˆ0.84930\sqrt{6} [6 \zeta(3) + 6 - \pi^2] / ( \pi \sqrt{18 + 9\pi^2 - \pi^4}) \approx 0.84930 in the limit as nβ†’βˆžn \rightarrow \infty. Although EnE_n has an expectation of 2 for all nn and HnH_n has expectation 2 in the limit as nβ†’βˆžn \rightarrow \infty, their limiting correlation is 0. The results contribute toward understanding features of the shapes of coalescent trees