Classification of some countable descendant-homogeneous digraphs
  For finite q, we classify the countable, descendant-homogeneous digraphs in which the descendant set of any vertex is a q-valent tree. We also give conditions on a rooted digraph G which allow us to construct a countable descendant-homogeneous digraph in which the descendant set of any vertex is isomorphic to G. 
Daniela Amato, David M. Evans, John K. Truss
math.CO math.GR