  The Rooted Maximum Leaf Outbranching problem consists in finding a spanning
directed tree rooted at some prescribed vertex of a digraph with the maximum
number of leaves. Its parameterized version asks if there exists such a tree
with at least $k$ leaves. We use the notion of $s-t$ numbering to exhibit
combinatorial bounds on the existence of spanning directed trees with many
leaves. These combinatorial bounds allow us to produce a constant factor
approximation algorithm for finding directed trees with many leaves, whereas
the best known approximation algorithm has a $\sqrt{OPT}$-factor. We also show
that Rooted Maximum Leaf Outbranching admits a quadratic kernel, improving over
the cubic kernel given by Fernau et al.
