We prove that finding a rooted subtree with at least k leaves in a directed graph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in strongly connected directed graphs. This settles completely an open question of Fellows and solves another one for strongly connected directed graphs. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: Every strongly connected digraph on n vertices with minimum in-degree at least 3, contains a rooted spanning tree with at least (n/2){1/5} leaves.
This is a joint work with Noga Alon, Fedor Fomin, Gregory Gutin and Michael
Krivelevich .