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

On Finding Directed Trees with Many Leaves