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CSE331_HW3

# CSE331_HW3 - 2200 k P(k 4.5 A proof by mathematical...

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3.27 If the recursive routine in Section 2.4 used to compute Fibonacci numbers is run for N=50, is stack space likely to run out? Why or why not? long fib( int n) { if (n<=1) return 1; else return fib(n-1) + fib(n-2); } It is likely to run out of space because for this recursive routine because as N gets larger the running time and the amount of space used grows exponentially. If this was not a stack and was an array using a loop there would never be a space issue unless N was extremely large. So basically it boils down to how stack’s elements are accessed and the way this particular function is run especially since it is recursive. 4.4 A proof by mathematical induction that P(n) filled nodes -> P(n+1) null nodes is true for every positive integer n consists of 2 steps. 1. Basis step. Show that proposition P(1) is true. 1 N N 2. Inductive step. For every positive k, the implication P(k) P(k+1) is true. P(k) for every k is proved by inductive hypothesis . [P(1) 2200 k (P(k) P(k+1))]

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Unformatted text preview: 2200 k P(k) 4.5 A proof by mathematical induction that the maximum number of nodes in a binary tree of height P(h) = 2 h+1-1 is true for all positive integers h. 1. Basis step. Show that proposition P(1) is true. P(1) = 2 1+1-1 = 3. As the following tree shows, the height is one because the root is not including and there are 3 nodes. 1 N N 2. Inductive step. For every positive k, the implication P(k) → P(k+1) is true. P(k) = 2 k+1-1 <=> P(k+1) = 2 (k+1)+1-1 2 k+1-1 <=> 2 (k+1)+1-1 (2 k + 2) -1 <=> (2 k + 2 2 ) -1 2 k + 2 <=> 2 k + 4 2<=> 4 2 1 <=> 2 1+1 2 k <=> 2 (k+1) 4.6 A proof by mathematical induction that P(n) filled nodes -> P(n+1) null nodes is true for every positive integer n consists of 2 steps. 1. Basis step. Show that proposition P(1) is true. 1 N N 2. Inductive step. For every positive k, the implication P(k) → P(k+1) is true. P(k) for every k is proved by inductive hypothesis . [P(1) ∧ 2200 k (P(k) → P(k+1))] → 2200 k P(k)...
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