Appears in collection : Schlumberger workshop on Topics in Applied Probability

This is joint work with Amaury Lambert (UPMC and Collège de France). We consider general branching processes where life lengths are i.i.d. with arbitrary distribution and births occur in a Poissonian manner. The corresponding genealogical trees are called splitting trees and can be characterized by a contour process with jumps, which a Levy process without negative jumps. This contour process allows to describe the genealogy of a population at a given time with a so-called coalescent point process. In this talk, we consider a supercritical population with Poissonian mutations within the infinite allele framework. The population at time t is partitioned into several "families" woth different alleles. We study the size of the largest families when t goes to infinity, depending whether the clonal process is subcritical, critical or supercritical. In particular, we are able to prove the convergence of the conveniently scaled point process of largest families.