Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees

The stationary birth-only, or Yule–Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or “ancestors”, and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of “birth” of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred.The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations.For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves.In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.

► Probabilities have been derived for root-to-leaf and leaf-to-leaf genealogical distances on rooted binary trees generated under the Yule model.
► For trees conditioned on the number n of tips, explicit expressions are given for these two distributions, and for their mean and variance in terms of n.
► For large n, both tend to become Poissonian, with parameter 2ln n for the first, and 4ln n for the second distribution.
► Non-conditioned distributions follow from an infinite hierarchy of differential equations in terms of (scaled) time, solvable only for the first distribution.
► Averages for two tree statistics were obtained using the methods developed in the paper: the Sackin index, and the sum over all leaf-to-leaf distances.