(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?
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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 $S$.