Let G be a nontrivial connected graph of order n and let k be an integer with 2≤k≤n. For a set S of k vertices of G, let κ(S) denote the maximum number 𝓁 of edge‐disjoint trees T
1,T
2,…,T
𝓁
in G such that V(T
i
)⋂V(T
j
)=S for every pair i,j of distinct integers with 1≤i,j≤𝓁. Chartrand et al. generalized the concept of connectivity as follows: The k‐connectivity, denoted by κ
k
(G), of G is defined by κ
k
(G)=min{κ(S)}, where the minimum is taken over all k‐subsets S of V(G). Thus κ
2(G)=κ(G), where κ(G) is the connectivity of G, for which there are polynomial‐time algorithms to solve it. This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k
1 and k
2, given a graph G and a k
1‐subset S of V(G), the problem of deciding whether G contains k
2 internally disjoint trees connecting S can be solved by a polynomial‐time algorithm. Then, we show that when k
1 is a fixed integer of at least 4, but k
2 is not a fixed integer, the problem turns out to be NP‐complete. On the other hand, when k
2 is a fixed integer of at least 2, but k
1 is not a fixed integer, we show that the problem also becomes NP‐complete.
