  We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament
contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving
the previously best known bound of $n+O(k^2)$ vertices to give a result tight
up to the value of $C$. Furthermore, we show that, for each $k$, there exists
$n_0$, such that, whenever $n\geqslant n_0$, any $(n+k-2)$-vertex tournament
contains a copy of every $n$-vertex oriented tree with at most $k$ leaves,
confirming a conjecture of Dross and Havet.
