Spanning tree with chosen leaves NP-Complete proof
The question you link to shows that the problem can be solved in polynomial time.  If the problem was NP-complete, then this would prove that P = NP.  Of course, it's a famous open problem to prove that P = NP or P != NP.  Therefore, you shouldn't expect anyone to know of any such proof.

Also, many computer scientists expect P != NP; if they're right, then the problem you list is not NP-complete.