We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph and a nonnegative integer , the goal is to construct a spanning out-tree of such that at least vertices in have the same out-degree as in . We show that this problem is W[1]-hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning out-tree such that at most vertices in have out-degrees that are different from that in . We show that this problem is fixed-parameter tractable and that it admits a problem kernel with at most vertices on strongly connected digraphs and vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time , where is the number of vertices in the input digraph.