  Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the
problem of finding in $D$ an out-branching with the minimum possible number of
leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved
that MinLOB is polynomial time solvable for acyclic digraphs which are exactly
the digraphs of directed path-width (DAG-width, directed tree-width,
respectively) 0. We investigate how much one can extend this polynomiality
result. We prove that already for digraphs of directed path-width (directed
tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand,
we show that for digraphs of restricted directed tree-width (directed
path-width, DAG-width, respectively) and a fixed integer $k$, the problem of
checking whether there is an out-branching with at most $k$ leaves is
polynomial time solvable.
