On the degree distribution of large tentacular Bienaym\'e-Galton-Watson trees

Abstract

We study Bienaym\'e-Galton-Watson trees conditioned to have $n$ vertices and $k_n$ leaves, where $k_n\to\infty$ while remaining negligible compared to $n$, i.e. $k_n=o(n)$. More precisely, we determine the asymptotic distribution of the outdegrees of these trees. We first show that the tree is asymptotically binary: the number of vertices with two children is asymptotically equal to the number of leaves, while almost all remaining vertices have exactly one child. Then, we identify the scales at which vertices with larger outdegrees emerge. For every $d\ge2$, the critical scale $k_n \sim cn^{(d-1)/d}$, $c>0$, is the threshold for the appearance of vertices with outdegree $d+1$. Below this scale, such vertices are absent with high probability; at the critical scale, their number converges to a Poisson distribution; above it, they satisfy a law of large numbers. Our proofs rely on the coding of Bienaym\'e-Galton-Watson trees by their {\L}ukasiewicz paths and asymptotic estimates for associated random walks.