  For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and
$|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq
V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if
$A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$
are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let
$\kappa_{S,r}(D)$ and $\lambda_{S,r}(D)$ be the maximum number of internally
disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized
$k$-vertex-strong connectivity of $D$ is defined as $$\kappa_k(D)= \min
\{\kappa_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ Similarly, the
generalized $k$-arc-strong connectivity of $D$ is defined as $$\lambda_k(D)=
\min \{\lambda_{S,r}(D)\mid S\subset V(D), |S|=k, r\in S\}.$$ The generalized
$k$-vertex-strong connectivity and generalized $k$-arc-strong connectivity are
also called directed tree connectivity which extends the well-established tree
connectivity on undirected graphs to directed graphs and could be seen as a
generalization of classical connectivity of digraphs.
  In this paper, we completely determine the complexity for both $\kappa_{S,
r}(D)$ and $\lambda_{S, r}(D)$ on general digraphs, symmetric digraphs and
Eulerian digraphs. In particular, among our results, we prove and use the
NP-completeness of 2-linkage problem restricted to Eulerian digraphs. We also
give sharp bounds and characterizations for the two parameters $\kappa_k(D)$
and $\lambda_k(D)$.
