  The question of shellability of complexes of directed trees was asked by R.
Stanley. D. Kozlov showed that the existence of a complete source in a directed
graph provides a shelling of its complex of directed trees. We will show that
this property gives a shelling that is straightforward in some sense. Among the
simplicial polytopes, only the crosspolytopes allow such a shelling.
Furthermore, we show that the complex of directed trees of a complete double
directed graph is a union of suitable spheres. We also investigate shellability
of the maximal pure skeleton of a complex of directed trees. Also, we prove
that is vertex-decomposable. For these complexes we describe the set of
generating facets.
