Part(a)
Base case: h=1 There is only one such tree with one leaf node and no full node. Hence the statement holds for base case.
Inductive step: h=k+1
Case 1: root is not a full node.
we assume it does not have a right child. In this case the number of full nodes and the the number of leaf nodes is equal to the tree which is rooted at at's left child. Since the height of that left subtree is k, by induction the difference should be 1.
Case 2: root is full node.
Total number of leaf nodes = number of leaf nodes in left + number of leaf nodes right subtree.
Total number of full nodes = Root (1) + the number of full nodes to its left and right.
Thus the difference remains 1.
Part (b)