(1) Given a connected graph G of p vertices. Is it possible to find randomly a tree T of G then assume the end vertices of T to be a set S having Steiner tree T. (2) Given a set S of k-vertices, is it possible to construct a graph G of p vertices(p>k) containing S such that G contains a Steiner tree T of S. Is the above problems are equivalent to the Steiner tree problem?
The questions are very unclear. I'll make some interpretations and give an answer. If my interpretations are incorrect, please edit your question to clarify.
(1) Suppose $G$ is a complete graph on $p=4$ vertices at the corners of a square. If $T$ is any subtree with 3 or 4 vertices then $T$ will not be a Steiner tree for its "end vertices".
(2) Given $S$, the Steiner tree $T$ of $S$ is itself a graph containing a Steiner tree of $S$.