  We consider a binary branching process structured by a stochastic trait that
evolves according to a diffusion process that triggers the branching events, in
the spirit of Kimmel's model of cell division with parasite infection. Based on
the observation of the trait at birth of the first n generations of the
process, we construct nonparametric estimator of the transition of the
associated bifurcating chain and study the parametric estimation of the
branching rate. In the limit, as n tends to infinity, we obtain asymptotic
efficiency in the parametric case and minimax optimality in the nonparametric
case.
