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The fractional arboricity of a digraph $D$, denoted by $\gamma(D)$, isdefined as $\gamma(D)= \max_{H \subseteq D, |V(H)| >1} \frac {|A(H)|}{|V(H)|-1}$. Frank in [Covering branching, Acta Scientiarum Mathematicarum(Szeged) 41 (1979), 77-81] proved that a digraph $D$ decomposes into $k$branchings, if and only if $\Delta^{-}(D) \leq k$ and $\gamma(D) \leq k$.  In this paper, we study digraph analogues for the Nine Dragon TreeConjecture. We conjecture that, for positive integers $k$ and $d$, if $D$ is adigraph with $\gamma(D) \leq k + \frac{d-k}{d+1}$ and $\Delta^{-}(D) \leq k+1$,then $D$ decomposes into $k + 1$ branchings $B_{1}, \ldots, B_{k}, B_{k+1}$with $\Delta^{+}(B_{k+1}) \leq d$. This conjecture, if true, is a refinement ofFrank's characterization. A series of acyclic bipartite digraphs is alsopresented to show the bound of $\gamma(D)$ given in the conjecture is bestpossible.  We prove our conjecture for the cases $d \leq k$. As more evidence to supportour conjecture, we prove that if $D$ is a digraph with the maximum averagedegree $mad(D)$ $\leq$ $2k + \frac{2(d-k)}{d+1}$ and $\Delta^{-}(D) \leq k+1$,then $D$ decomposes into $k + 1$ pseudo-branchings $C_{1}, \ldots, C_{k},C_{k+1}$ with $\Delta^{+}(C_{k+1}) \leq d$.

Digraph analogues for the Nine Dragon Tree Conjecture