Trees and Treelike 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 + 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 collections of at most O(n^0.99) pairwise vertex-disjoint cycles and subdivisions of graphs H with |H| < exp(√O(log n)) in which each edge is subdivided at least once.