Semidegree threshold for spanning trees in oriented graphs

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