  In 2001, Koml\'os, S\'ark\"ozy and Szemer\'edi proved that, for each
$\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every
$n$-vertex graph with minimum degree at least $(1/2+\alpha)n$ contains a copy
of every $n$-vertex tree with maximum degree at most $cn/\log n$. We prove the
corresponding result for directed graphs. That is, for each $\alpha>0$, there
is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex
directed graph with minimum semi-degree at least $(1/2+\alpha)n$ contains a
copy of every $n$-vertex oriented tree whose underlying maximum degree is at
most $cn/\log n$.
  As with Koml\'os, S\'ark\"ozy and Szemer\'edi's theorem, this is tight up to
the value of $c$. Our result improves a recent result of Mycroft and Naia,
which requires the oriented trees to have underlying maximum degree at most
$\Delta$, for any constant $\Delta$ and sufficiently large $n$. In contrast to
the previous work on spanning trees in dense directed or undirected graphs, our
methods do not use Szemer\'edi's regularity lemma.
