  We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput.
Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about
finding disjoint "balanced" spanning rooted substructures in graphs and
digraphs, which generalize classic packing problems. Namely, given a positive
integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem
of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem
of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these
problems are FPT with parameter $k$, improving on existing XP algorithms. The
latter of these results answers a question of Bang-Jensen, Havet, and Yeo
[Discret. Appl. Math. 2016]. Further, given an integer $k$, a graph $G=(V,E)$,
and $r \in V$, we consider the problem of finding two arc-disjoint $(r,k)$-safe
spanning trees. We show that this problem is also FPT with parameter $k$, again
improving on a previous XP algorithm. Our main technical contribution is to
prove that the existence of such spanning substructures is equivalent to the
existence of substructures with size and maximum (out-)degree both bounded by a
(linear or quadratic) function of $k$, which may be of independent interest.
