Scaling limits of slim and fat trees

Abstract

We consider Galton--Watson trees conditioned on both the total number of vertices $n$ and the number of leaves $k$. The focus is on the case in which both $k$ and $n$ grow to infinity and $k = \alpha n + O(1)$, with $\alpha \in (0, 1)$. Assuming the exponential decay of the offspring distribution, we show that the rescaled random tree converges in distribution to Aldous' Continuum Random Tree with respect to the Gromov--Hausdorff topology. The scaling depends on a parameter $\sigma^\ast$ which we calculate explicitly. Additionally, we compute the limit for the degree sequences of these random trees.