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.

37 pages, 2 figures

Cited by in corpus (1)

• Conditioning Bienaym{é}-Galton-Watson trees to have large sub-populations