Answer the following questions so as to justify Proposition 8.8.

a. What is the minimum number of external nodes for a proper binary tree with height h? Justify your answer.

b. What is the maximum number of external nodes for a proper binary tree with height h? Justify your answer.

c. Let T be a proper binary tree with height h and n nodes. Show that log(n+1)−1 ≤ h ≤ (n−1)/2.

d. For which values of n and h can the above lower and upper bounds on h be attained with equality?