Counting oriented trees in digraphs with large minimum semidegree

Abstract

Let T be an oriented tree on n vertices with maximum degree at most $e^{o\left(\sqrt{\log n}\right)}$. If G is a digraph on n vertices with minimum semidegree ${\delta }^0\left(G\right)\ge (\frac{1}{2}+o(1\left)\right)n$, then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree $o(n/\log n)$). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least $|\mathrm{Aut}(T){|}^{-1}{(\frac{1}{2}-o(1\left)\right)}^{n}n!$ copies of T, which is optimal. This implies the analogous result in the undirected case.