Invariance principles for Galton–Watson trees conditioned on the number of leaves

We are interested in the asymptotic behavior of critical Galton–Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton–Watson tree having nn leaves. Second, we let tntn be a critical Galton–Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly nn leaves. We show that the rescaled Lukasiewicz path and contour function of tntn converge respectively to XexcXexc and Hexc, where XexcXexc is the normalized excursion of a strictly stable spectrally positive Lévy process and Hexc is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton–Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton–Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.