For a digraph $$D=(V(D), A(D))$$ , and a set $$S\subseteq V(D)$$ with $$r\in S$$ and $$|S|\ge 2$$ , 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 $$\begin{aligned} \kappa _k(D)= \min \{\kappa _{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}. \end{aligned}$$ Similarly, the generalized k-arc-strong connectivity of D is defined as $$\begin{aligned} \lambda _k(D)= \min \{\lambda _{S,r}(D)\mid S\subseteq V(D), |S|=k, r\in S\}. \end{aligned}$$ The generalized k-vertex-strong connectivity and generalized k-arc-strong connectivity are also called directed tree connectivity which could be seen as a generalization of classical connectivity of digraphs and a natural extension of undirected tree connectivity. A digraph $$D=(V(D), A(D))$$ is called minimally generalized $$(k, \ell )$$ -vertex (respectively, arc)-strongly connected if $$\kappa _k(D)\ge \ell $$ (respectively, $$\lambda _k(D)\ge \ell $$ ) but for any arc $$e\in A(D)$$ , $$\kappa _k(D-e)\le \ell -1$$ (respectively, $$\lambda _k(D-e)\le \ell -1$$ ). In this paper, we study the minimally generalized $$(k, \ell )$$ -vertex (respectively, arc)-strongly connected digraphs. We compute the minimum and maximum sizes of these digraphs and give characterizations of such digraphs for some pairs of k and $$\ell $$ .