  In this paper, we study the problem of finding a minimum weight spanning tree
that contains each vertex in a given subset $V_{\rm NT}$ of vertices as an
internal vertex. This problem, called Minimum Weight Non-Terminal Spanning
Tree, includes $s$-$t$ Hamiltonian Path as a special case, and hence it is
NP-hard. In this paper, we first observe that Non-Terminal Spanning Tree, the
unweighted counterpart of Minimum Weight Non-Terminal Spanning Tree, is already
NP-hard on some special graph classes. Moreover, it is W[1]-hard when
parameterized by clique-width. In contrast, we give a $3k$-vertex kernel and
$O^*(2^k)$-time algorithm, where $k$ is the size of non-terminal set $V_{\rm
NT}$. The latter algorithm can be extended to Minimum Weight Non-Terminal
Spanning Tree with the restriction that each edge has a polynomially bounded
integral weight. We also show that Minimum Weight Non-Terminal Spanning Tree is
fixed-parameter tractable parameterized by the number of edges in the subgraph
induced by the non-terminal set $V_{\rm NT}$, extending the fixed-parameter
tractability of Minimum Weight Non-Terminal Spanning Tree to the general case.
Finally, we give several results for structural parameterization.
