NP-completeness of a spanning tree problem

I was reviewing some NP-complete problems on this site, and I meet one interesting problem from

NP completeness proof of a spanning tree problem

In this problem, I am interested in the original problem, which the leaf set is precisely $S$. The author said that he can prove this by reducing it to the Hamiltonian path. However, I still cannot figure it out. Could anybody help me with this in details?

• Hint: What can you say about a tree with only two leaves? What if it's a spanning tree? Apr 16 '12 at 8:36
• Welcome! What have you tried?
– Raphael
Apr 16 '12 at 18:57
• Thank you! I know that the H-path is a kind of spanning tree, and with this fact, I almost done. But, for the remain part, I don't know how to make a correspondence of the H-path problem to this problem, to make the leaves exactly $S$. I mean "leaves are precisely in $S$" is too tricky for me to create a correspondence...Thank you! Apr 19 '12 at 19:45