If a digraph has no oriented 4-cycles, minimum semidegree at least k/2, and at least one vertex with outdegree and indegree at least k, then it contains every oriented tree with k arcs.
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Oriented Trees in Digraphs without Oriented $4$-cycles
If a digraph has no oriented 4-cycles, minimum semidegree at least k/2, and at least one vertex with outdegree and indegree at least k, then it contains every oriented tree with k arcs.