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 n leaves. Second, we let ^tn be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly n leaves. We show that the rescaled Lukasiewicz path and contour function of ^tn converge respectively to ^Xexc and ^Hexc, where ^Xexc 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.