Hints: Problem 2.25: Do induction on number of vertices. The hard part is to figure out how to use the induction hypothesis. If one of the subtrees contains only one vertex then the problem is easy. Otherwise you have to work harder: Suppose all subtrees contain at least 2 nodes. When going from n+1 to n vertices, try removing a leaf. Note that after removing the leaf, all subtrees are still non-empty. Now try to use the induction hypothesis.
Problem 2.31: Prove simultaneously that:
F(2n+1) = F(n)^2 + F(n+1)^2 F(2n) = F(n) F(n+1) + F(n) F(n-1)Note that the proof by induction will need cases depending on whether you are proving the case for m of the form 2n or 2n+1.