Semidegree threshold for spanning trees in oriented graphs

We show that for all $\gamma >0$ and $\mathrm{\Delta }\in \mathbb{N}$, there is some $n_0$ such that, if $n\ge n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3\mathrm{/}8+\gamma )n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $\mathrm{\Delta }$. This is asymptotically best possible.