Classification of some countable descendant-homogeneous digraphs

For finite qq, we classify the countable, descendant-homogeneous digraphs in which the descendant set of any vertex is a qq-valent tree. We also give conditions on a rooted digraph ΓΓ which allows us to construct a countable descendant-homogeneous digraph in which the descendant set of any vertex is isomorphic to ΓΓ.