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 L\'evy 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 $N_k(n)$, depending on the position of $k$ relative to a model-dependent critical value.