A compendium of covariances and correlation coefficients of coalescent tree properties

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: H_n, L_n, E_n, I_n, B_n, and T_k for a sample of size n, with 2≤k≤n. We report high correlations among H_n, L_n, I_n, and B_n, 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 E_n has expectation 2 for all n and H_n has expectation 2 in the n→∞ limit, their limiting correlation is 0. The results contribute toward understanding features of the shapes of coalescent trees.