  A spanning subgraph $F$ of a graph $G$ is called {\em perfect} if $F$ is a
forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of
$F$ is an induced subgraph of $G$. Alex Scott (Graphs \& Combin., 2001) proved
that every connected graph $G$ contains a perfect forest if and only if $G$ has
an even number of vertices. We consider four generalizations to directed graphs
of the concept of a perfect forest. While the problem of existence of the most
straightforward one is NP-hard, for the three others this problem is
polynomial-time solvable. Moreover, every digraph with only one strong
component contains a directed forest of each of these three generalization
types. One of our results extends Scott's theorem to digraphs in a non-trivial
way.
