  We provide the directed counterpart of a slight extension of Katoh and
Tanigawa's result on rooted-tree decompositions with matroid constraints. Our
result characterises digraphs having a packing of arborescences with matroid
constraints. It is a proper extension of Edmonds' result on packing of spanning
arborescences and implies - using a general orientation result of Frank - the
above result of Katoh and Tanigawa. We also give a complete description of the
convex hull of the incidence vectors of the basic packings of arborescences and
prove that the mimimum cost version of the problem can be solved in polynomial
time.
