  In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are
given a directed graph $D$ with a designated root $r$ and a nonnegative integer
$k$. The question is to determine the existence of an outbranching rooted at
$r$ that has at least $k$ leaves, or at least $k$ internal vertices,
respectively. Both these problems were intensively studied from the points of
view of parameterized complexity and kernelization, and in particular for both
of them kernels with $O(k^2)$ vertices are known on general graphs. In this
work we show that $k$-Leaf Out-Branching admits a kernel with $O(k)$ vertices
on $\mathcal{H}$-minor-free graphs, for any fixed family of graphs
$\mathcal{H}$, whereas $k$-Internal Out-Branching admits a kernel with $O(k)$
vertices on any graph class of bounded expansion.
