Ancestral diversity in fragmentation trees

In a deterministic or random tree, a notion of ancestral diversity can be defined as follows. Sample independently n groups of k leaves and count the number N n ( k ) of distinct most recent common ancestors of each of the groups. As n becomes large, the asymptotic behavior of N n ( k ) depends of course on the structure of the tree. Motivated by the study of the edge density in the Brownian co-graphon, Chapuy recently considered this problem in the case where k = 2 and where the tree is the Brownian continuum random tree. We vastly extend this framework by considering general values of k and general fragmentation trees, which include some prominent examples such as stable Lévy trees and idealized models of phylogenetic trees. Other natural ancestral statistics are also considered. For a given tree model, we identify a phase transition-like phenomenon, with different asymptotic regimes for N n ( k ), depending on the position of k relative to a model-dependent critical value.