Random trees conditioned on the number of vertices and leaves

I will talk about Galton-Watson trees conditioned on both the total number of vertices $n$ and the number of leaves $k$. Both $k$ and $n$ are assumed to grow to infinity and $k = alpha n + O(1)$, with $alpha in (0, 1)$. Assuming the exponential decay of the offspring distribution, I show that the rescaled random tree converges in distribution to Aldous’ Continuum Random Tree with respect to the Gromov-Hausdorff topology. The rescaling depends on a parameter $sigma^2$ which can be calculated explicitly. Additionally, I will describe the limit of the degree sequence for the conditioned trees.