Scaling limits of slim and fat trees

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.