Trees and tree-like structures in dense digraphs

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 Komlós, Sárközy and Szemerédi. 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.