A compendium of covariances and correlation coefficients of coalescent tree properties


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

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