Can't you see that you confused n and h? Just to remark, "external nodes" are also (in my experience, more often) called "leaves".t is a proper binary tree with n nodes and h is the height... The number of external nodes in T is at least h+1 and at most 2^h. or h>=h+1 and h<=2^h.
When people have difficulties with induction, I think it is extremely important to write P(h) explicitly and to make sure that it is a proposition, not a number. This may not be the issue here, but it makes everything else easier.So i let P(h) be that statement.