Clade Size Statistics Under Ford’s α -Model

Given a labeled tree topology t of n taxa, consider a population P of k leaves chosen among those of t . The clade of P is the minimal subtree P ^ of t containing P , and its size | P ^ | is provided by the number of leaves in the clade. We study distributive properties of the clade size variable | P ^ | considered over labeled topologies of size n generated at random in the framework of Ford’s α -model. Under this model, starting from the one-taxon labeled topology, a random labeled topology is produced iteratively by a sequence of α -insertions, each of which adds a pendant edge to either a pendant or internal edge of a labeled topology, with a probability that depends on the parameter α ∈ [ 0 , 1 ] . Different values of α determine different probability distributions over the set of labeled topologies of given size n , with the special cases α = 0 and α = 1 / 2 respectively corresponding to the Yule and uniform distributions. In the first part of the manuscript, we consider a labeled topology t of size n generated by a sequence of random α -insertions starting from a fixed labeled topology t ∗ of given size k , and determine the probability mass function, mean, and variance of the clade size | P ^ | in t when P is chosen as the set of leaves of t inherited from t ∗ . In the second part of the paper, we calculate the probability that a set P of k leaves chosen at random in a Ford-distributed labeled topology of size n is monophyletic, that is, the probability that | P ^ | = k . Our investigations extend previous results on clade size statistics obtained for Yule and uniformly distributed labeled topologies.