A compendium of covariances and correlation coefficients of coalescent tree properties.

ABSTRACT

Gene genealogies are frequently studied by measuring properties such as their height (H), length (L), sum of external branches (E), sum of internal branches (I), and mean of their two basal branches (B), and the coalescence times that contribute to the other genealogical features (T). 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 Hn, Ln, En, In, Bn, and Tk for a sample of size n, with 2≤k≤n. We report high correlations among Hn, Ln, In, and Bn, with all pairwise correlations of these quantities having values greater than or equal to 6[6ζ(3)+6-π2]/(π18+9π2-π4)≈0.84930 in the limit as n→∞, where ζ is the Riemann zeta function. Although En has expectation 2 for all n and Hn has expectation 2 in the n→∞ limit, their limiting correlation is 0. The results contribute toward understanding features of the shapes of coalescent trees.