  The {\sc Directed Maximum Leaf Out-Branching} problem is to find an
out-branching (i.e. a rooted oriented spanning tree) in a given digraph with
the maximum number of leaves. In this paper, we improve known parameterized
algorithms and combinatorial bounds on the number of leaves in out-branchings.
We show that
  \begin{itemize} \item every strongly connected digraph $D$ of order $n$ with
minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$
leaves; \item if a strongly connected digraph $D$ does not contain an
out-branching with $k$ leaves, then the pathwidth of its underlying graph is
$O(k\log k)$; \item it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$
whether a strongly connected digraph on $n$ vertices has an out-branching with
at least $k$ leaves. \end{itemize}
  All improvements use properties of extremal structures obtained after
applying local search and of some out-branching decompositions.
