  The aim of this paper is to reveal the discrete convexity of the minimum-cost
packings of arborescences and branchings. We first prove that the minimum-cost
packings of disjoint $k$ branchings (minimum-cost $k$-branchings) induce an
$\mathrm{M}^\natural$-convex function defined on the integer vectors on the
vertex set. The proof is based on a theorem on packing disjoint $k$-branchings,
which extends Edmonds' disjoint branchings theorem and is of independent
interest. We then show the $\mathrm{M}$-convexity of the minimum-cost
$k$-arborescences, which provides a short proof for a theorem of Bern\'ath and
Kir\'aly (SODA 2016) stating that the root vectors of the minimum-cost
$k$-arborescences form a base polyhedron of a submodular function. Finally,
building upon the $\mathrm{M}^\natural$-convexity of $k$-branchings, we present
a new problem of minimum-cost root location of a $k$-branching, and show that
it can be solved in polynomial time if the opening cost function is
$\mathrm{M}^\natural$-convex.
