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1 1. Prove: A full m-ary tree with i internal vertices contains n = mi + 1 vertices. 2. For a full m-ary tree with n vertices, i internal vertices, and l leaves, prove: (i) i = (n 1)/m and l = [(m 1)n + 1]/m (ii) n = mi + 1 vertices and l = (m 1)i + 1 (iii) n = (ml 1)/(m 1) and i = (l 1)/(m 1) 3-8. Refer to this rooted tree: 3. According to universal addressing, what is the address of vertex d? 4. According to universal addressing, what is the address of vertex f? 5. According to universal addressing, what is the address of vertex j? 6. Write the preorder traversal. 7. Write the postorder traversal. 8. Write the inorder traversal Refer to this rooted tree:

2 9. According to universal addressing, what is the address of vertex g? 10. According to universal addressing, what is the address of vertex l? 11. According to universal addressing, what is the address of vertex o? ] 12. Write the preorder traversal. 13. Write the postorder traversal. 14. Write the inorder traversal. 15. Draw the binary tree for the expression (5 (7 + 3)) (8 2) 16. Referring to #15, write the expression in prefix notation. 17. Prove: On a tree, there is a unique simple path between any two vertices. 18. Prove: A tree with n vertices has exactly n 1 edges. 19. Draw the binary tree for the expression (8 3) + ((6 + 2) 5). 20. Referring to #19, write the expression in prefix and postfix notation. 21. Draw the binary tree for the expression (5 (3 + (2 4))) (2 + 9) 22. Referring to #21, write the expression prefix and postfix notation. 23. Use strong induction to prove: The number of leaves on an m-ary tree of height h is at most m h, for h 1 (the case where h=0 is trivial; for the basis, prove P(1).). 24. Evaluate the postfix expression: Draw the binary tree for #24.

3 26. Referring to #24, write the expression in prefix notation. 27. Refer to this graph: A. Suppose we perform a breadth first search rooted at c. If the first two edges are {c, a} and {a, f }, what is the next edge? B. Suppose we perform a breadth first search rooted at e. If the first three edges are {e, h}, {e, n} and {h, g}, what is the next edge? C. Suppose we perform a depth first search rooted at d, starting with the path d, b, a, f. What edge is added next? D. Suppose we perform a breadth first search rooted at d. If the first two edges are {d, f}, and {d, n}, what is the next edge? E. Suppose we perform a depth first search rooted at b, starting with the path b, n, e, h, i. What edge is added next? F. Suppose we perform a depth first search rooted at e, starting with the path e, n, b, d, f, a, c. What edge is added next?

4 28. Refer to this graph: A. Suppose we perform a depth first search rooted at f, starting with the path f, d, b, a. Continuing the search, what will the final edge added to the spanning tree? B. Suppose we perform a depth first search rooted at e, employing the path e, b, d, f Continuing the search, what will then next vertex added to the spanning tree? C. Suppose we perform a breadth first search rooted at c, starting with the edges {c, h}, {c, b}. What will be the next edge? D. Suppose we perform a breadth first search rooted at f, starting with the edges {f, d}, {d, e}. What will be the next edge? 29. Use Boolean identities, and algebra, to prove x y + y + z = y z + x ( ) Justify each step in the proof. Do not skip or combine steps. 30. Use Boolean identities, and algebra, to prove x xy + zy ( ) = x 31. T F The set { +, } is functionally complete. 32. Write a Boolean expression that is equivalent to xyz + yz, but without any products. 33. T F The set {, } is functionally complete. 34. Write a Boolean expression that is equivalent to xyz + yz, but without any sums. 35. T F If the set {, } is functionally complete, then the set { } is functionally complete. 36. T F If the set { +, } is functionally complete, then the set {} is functionally complete. 37. Use algebra (not a truth table) to derive the sum of products expansion (aka disjunctive normal form) ( ) y + z ( ) = xz + x + yz ( ). You don t have to justify the steps. You may skip or combine steps. for F x, y,z You must write the answer using lexicographic order

5 38. Use algebra (not a truth table) to derive the sum of products expansion (aka disjunctive normal form) for F ( x, y,z) = x( y + z)+ yz. You don t have to justify the steps. You may skip or combine steps. You must write the answer using lexicographic order. 39. Use algebra (not a truth table) to derive the sum of products expansion (aka disjunctive normal form) for F ( x, y,z) = ( x + y) yz. You don t have to justify the steps. You may skip or combine steps. You must write the answer using lexicographic order. 40. Use Boolean identities, and algebra, to prove ( y + x) x + y =1. Justify each step of the proof. Do not skip or combine steps. 41. Use algebra to derive the sum of products expansion (aka disjunctive normal form) for ( ) ( ) = x + z + y x + z F x, y,z You may skip/combine steps. You don t have to write the justifications. Your answer must be written using lexicographic order 42. Use algebra, not a truth table, to derive the sum of products expansion for F ( x, y, x) = z( y + x) You may skip/combine steps. You do not have to justify the steps. You must write your answer using lexicographic order. 43. Find a Boolean function equivalent to ( x + y) ( yz) but without any sums. 44. Find a Boolean function equivalent to ( x + y) ( yz) but without any products. 45. Find the dual to ( x + y) ( yz) 46. Using, as needed, the Identity, Complements, Commutative, Associative and Distributive laws as axioms, prove this Idempotent law: x x = x. 47. Using, as needed, the Identity, Complements, Commutative, Associative and Distributive laws as axioms, prove this Idempotent law: x + x = x. 48. Having proven the Idempotent laws (Exercises 22 and 23 above), use them as needed, along with the Identity, Complements, Commutative, Associative and Distributive laws as axioms, to prove this Domination law: x 0 = Having proven the Idempotent laws (Exercises 22 and 23 above), use them as needed, along with the Identity, Complements, Commutative, Associative and Distributive laws as axioms, to prove this Domination law: x +1 = 1

6 50. Having proven the Idempotent and Domination laws (Exercises above), use them as needed, along with the Identity, Complements, Commutative, Associative and Distributive laws as axioms, to prove this Absorption law: x + xy = x 51. Having proven the Idempotent and Domination laws (Exercises above), use them as needed, along with the Identity, Complements, Commutative, Associative and Distributive laws as axioms, to ( ) = x prove this Absorption law: x x + y

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