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: Hnβ, Lnβ, Enβ, Inβ, Bnβ, and Tkβ for a sample
of size n, with 2β€kβ€n. We report high correlations among Hnβ,
Lnβ, Inβ, and Bnβ, with all pairwise correlations of these quantities
having values greater than or equal to 6β[6ΞΆ(3)+6βΟ2]/(Ο18+9Ο2βΟ4β)β0.84930 in the limit as nββ. Although Enβ has an expectation of 2 for all n and Hnβ has
expectation 2 in the limit as nββ, their limiting
correlation is 0. The results contribute toward understanding features of the
shapes of coalescent trees