Given a graph G=(V,E) where V and E are a vertex and an edge set, respectively, specified with a subset V_NT of vertices called a non-terminal set, the spanning tree with non-terminal set V_NT is a connected and acyclic spanning subgraph of G that contains all the vertices of V where each vertex in a non-terminal set is not a leaf. In the case where each edge has the weight of a nonnegative integer, the problem of finding a minimum spanning tree with a non-terminal set V_NT of G was known to be NP-hard. However, the complexity of finding a spanning tree on general graphs where each edge has the weight of one was unknown. In this paper, we consider this problem and first show that it is NP-hard even if each edge has the weight of one on general graphs. We also show that if G is a cograph then finding a spanning tree with a non-terminal set V_NT of G is linearly solvable when each edge has the weight of one.