# hw06 - T(4.3 Deﬁnition 4 analogously to how it is done...

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Math 55: Discrete Mathematics, Fall 2008 Reading and Homework Assignment 6 Reading: Lectures 16-17: 4.3; section at end of 4.4 on Merge Sort Lectures 17-18: 5.1 Homework (due Monday, 10/13): Odd-numbered self-checking exercises: 4.3: 1(d), 5(a,b,e), 13, 17, 39 4.4: 47 5.1: 15, 19, 29, 39 Problems to hand in: 4.3: 6(b), 12, 30, Ch. 4 Suppl. Ex. 18 [hint: determine f n (mod 3) for every n ] 4.4: 48(a,b). For each problem, either ﬁnd an algorithm that always uses fewer than the worst-case m + n - 1 steps required by 4.4, Algorithm 10, or show that no such algorithm is possible. 5.1: 16 [hint: subtract], 20(a-f), 28, 38, 42, 54, 58 (A) We deﬁne the height h ( T ) of a rooted tree
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Unformatted text preview: T (4.3, Deﬁnition 4) analogously to how it is done for full binary trees in 4.3, Deﬁnition 7, by replacing the maximum over T 1 ,T 2 in the recursive step by the maximum over all the con-stituent trees T 1 ,...,T n . We deﬁne the set of leaves recursively, analogously to the preamble to 4.3, Exercise 44, to be { r } in the case that T consists only of a root r , and otherwise to be the union of the sets of leaves of the trees T 1 ,...,T n . Let n ( T ) denotes the number of nodes in T , and l ( T ) the number of leaves. Prove that n ( T ) ≤ l ( T ) h ( T ) + 1 for every rooted tree T ....
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