We consider a Galton-Watson tree whose birth distribution depends on the hidden type of nodes: normal or special. Every special node gives birth to one special child and a number of normal children whose descendance will be normal. Even in such a very structured two-type population, our ability to distinguish the two types and estimate their birth distribution is constrained by a trade-off between the growth-rate of the population and the similarity of the two birth distributions. Indeed, if the growth-rate is too large, large deviations events are likely to be observed in the sampling of the normal individuals preventing us to distinguish them from special ones. The talk will be illustrated by numerical simulations and asymptotic goodness-of-fit tests for surviving subcritical Galton-Watson trees. Joint work with Benoît Henry.
