Figure 10.13 A tree

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CHAP. 10]

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Figure 10.14 A tree

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Which traversals always visit: a. the root first b. the left-most node first c. the root last d. the right-most node last The level order traversal follows the pattern as reading a page of English text: left-to-right, row-by-row. Which traversal algorithm follows the pattern of reading vertical columns from left to right Which traversal algorithm is used in the call tree for the solution to Problem 9.32 on page 184

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10.1 10.2 Prove Theorem 10.1 on page 187. Prove Theorem 10.2 on page 188.

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10.3 10.4 10.5 10.6 Prove Corollary 10.1 on page 188. Prove Corollary 10.2 on page 188.

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Derive the formula for the path length of a full tree of order d and height h. The St. Petersburg Paradox is a betting strategy that seems to guarantee a win. It can be applied to any binomial game in which a win or lose are equally likely on each trial and in which the amount bet on each trial may vary. For example, in a coin-flipping game, bettors may bet any number of dollars on each flip, and they will win what they bet if a head comes up, and they will lose what they bet if a tail comes up. The St. Petersburg strategy is to continue playing until a head comes up, and to double your bet each time it doesn t. For example, the sequence of tosses is {T, T, T, H}, then the bettor will have bet $1 and lost, then $2 and lost, then $4 and lost, then $8 and won, ending up with a net win of $1 + $2 + $4 + $8 = $1. Since a head has to come up eventually, the bettor is guaranteed to win $1, no matter how many coin flips it takes. Draw the transition diagram for this strategy showing the bettor s winnings at each stage of play. Then explain the flaw in this strategy. Some people play the game of craps allowing 3 to be a possible point. In this version, player Y wins on the first toss only if it comes up 2 or 12. Use a transition diagram to analyze this version of the game and compute the probability that X wins. Seven coins that appear identical are to be tested to determine which one of them is counterfeit. The only feature that distinguishes the counterfeit coin is that it weighs less than the legitimate coins. The only available test is to weigh one subset of the coins against another. How should the subsets be chosen to find the counterfeit (See Example 10.3 on page 188.)

Answers to Review Questions

10.1 In the Java inheritance tree: a. The size of the tree in Java 1.3 is 1730. b. The Object class is at the root of the tree. c. A final class is a leaf node in the Java inheritance tree. a. True. b. False: It s one more because the root of the subtree is in the subtree but is not a descendant of itself. c. True d. False e. True f. True g. False h. True i. True j. True k. False l. False m. True n. True a. The leaf nodes are L, M, N, H, O, P, Q; the children of node C are G and H; node F has depth 2; the nodes at 3 three are L, M, N, O, P, and Q; the height of the tree is 3; the order of the tree is 4. b. The leaf nodes are C, E, G, O, P, Q, R, and S; node C has no children; node F has depth 2; the nodes at level 3 are L, M, N, and O; the height of the tree is 4; the order of the tree is 4.

CHAP. 10]