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c. The leaf nodes are C, E, G, J, L, N, O, P, W, Y, and Z; node C has no children; node F has depth 2; the nodes at level 3 are H, J, and K; the height of the tree is 9; the order of the tree is 3. d. The leaf nodes are G, H, K, L, N, O, P, Q, R, S, and T; the only child node C has is node E; node F has depth 3; the nodes at level 3 are F, G, H, and J; the height of the tree is 5; the order is 5. e. The leaf nodes are D, E, L, N, P, Q, R, S, and T; node C has no children; node F has depth 1; the nodes at level 3 are K, L, M, N, and O; the height of the tree is 4; the order of the tree is 5. 10.4 a. b. c. d. a. b. c. d. The ancestors of F are C and A The descendants of F are I, K, and L. The nodes in the subtree rooted at F are F, I, K, and L. The leaf nodes are D, H, J, K, and L. (35 1)/2 = 121 nodes (44 1)/3 = 85 nodes (105 1)/9 = 11,111 nodes (411 1)/3 = 1,398,101 nodes

a. Level order: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o. b. Preorder: a, b, e, i, j, c, f, k, g, h, l, m, n, o, d. c. Postorder: i, j, e, b, k, f, g, l, m, n, o, h, c, d, a. a. b. c. d. The level order and the preorder traversals always visit the root first. The postorder traversal always visits the left-most node first. The postorder traversal always visits the root last. The preorder traversal always visits the right-most node last.

10.8 10.9

widening do not change output per worker; however, capital additions that are capital deepening increase output per worker When there is no change in the labor force, capital additions result in diminishing returns and have decreased rates of return Capital additions cease and the economy reaches a steady state when the rate of return from capital additions equals the economy s real rate of interest Since there is a limit to capital deepening when there is no change in technology, there must also be a limit to output per worker and therefore to the economy s standard of living An economy s steady-state position can be pushed to a higher level of output per worker by an increase in its rate of saving, by improved technology, and/or by better education of its population Productivity is measured by dividing real GDP by the total number of hours worked by labor.

The preorder traversal follows the pattern of reading by column from left to right. The preorder traversal is used in Problem 9.32 on page 184.

101 Proof of Theorem 101 on page 187: If there were no path from a given node x to the root of the tree, then the definition of tree would be violated, because to be an element of the tree, x must be the root of some subtree If there were more than one path from x back to the root, then x would be an element of more than one distinct subtree That also violates the definition of tree, which requires subtrees to be disjoint Proof of Theorem 102 on page 188: If the tree is empty, then its height is h = 1 and the number of nodes n = 0 In that case, the formula is correct: n = (d (h)+1 1)/(d 1) = (d ( 1)+1 1)/(d 1) = (d 0 1)/(d 1) = (1 1)/(d 1) = 0.

If the tree is a singleton, then its height is h = 0 and the number of nodes n = 1 In that case, the formula is again correct: n = (d (h)+1 1)/(d 1) = (d (0)+1 1)/(d 1) = (d 1)/(d 1) = 1 Now assume that the formula is correct for any full tree of height h 1, where h 0 Let T be the full tree of height h Then by definition, T consists of a root node and a set of d subtrees And since T is full, each of its d subtrees has height h 1 Therefore, by the inductive hypothesis, the number of nodes in each subtree is nS = (d (h 1)+1 1)/(d 1) = (d h 1)/(d 1).

Thus, the total number of nodes in T is n = 1 + d nS dh 1 = 1 + d ------------d 1 d 1 dh + 1 d = ----------- + -------------------d 1 d 1.

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