Approximations to the expectations and variances of ratios of tree properties under the coalescent

Properties of gene genealogies such as tree height (H), total branch length (L), total lengths of external (E) and internal (I) branches, mean length of basal branches (B), and the underlying coalescence times (T) can be used to study population-genetic processes and to develop statistical tests of population-genetic models. Uses of tree features in statistical tests often rely on predictions that depend on pairwise relationships among such features. For genealogies under the coalescent, we provide exact expressions for Taylor approximations to expected values and variances of ratios [Formula: see text] , for all 15 pairs among the variables [Formula: see text] , considering n leaves and [Formula: see text]. For expected values of the ratios, the approximations match closely with empirical simulation-based values. The approximations to the variances are not as accurate, but they generally match simulations in their trends as n increases. Although E(n) has expectation 2 and H(n) has expectation 2 in the limit as [Formula: see text] , the approximation to the limiting expectation for [Formula: see text] is not 1, instead equaling [Formula: see text]. The new approximations augment fundamental results in coalescent theory on the shapes of genealogical trees.