  We prove that every oriented tree on $n$ vertices with bounded maximum degree
appears as a spanning subdigraph of every directed graph on $n$ vertices with
minimum semidegree at least $n/2+\mathrm{o}(n)$. This can be seen as a directed
graph analogue of a well-known theorem of Koml\'os, S\'ark\"ozy and
Szemer\'edi. Our result for trees follows from a more general result, allowing
the embedding of arbitrary orientations of a much wider class of spanning
"tree-like" structures, such as a collection of at most $\mathrm{o}(n^{1/4})$
vertex-disjoint cycles and subdivisions of graphs $H$ with $|H|< n^{(\log
n)^{-1/2}}$ in which each edge is subdivided at least once.
