lec13-recursive - tree with the root r Each T i is called a...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 13 Recursive structures February 15 Reading: Rosen [4.3] February 15 Recursive structures, Page 1
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13.1 More inductions Critical steps in applying induction: The number to do induction on. Reduce k + 1 case to k case. Example. Suppose a m,n is defined as follows: a 0 , 0 = 0 , a m,n = a m - 1 ,n + 1 if n = 0 and m > 0 a m,n - 1 + n otherwise. Prove that a m,n = m + n ( n + 1) / 2 for all m, n N . Proof: Induction on m + n . February 15 Recursive structures, Page 2
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13.2 Fibonacci numbers f 0 = 0, f 1 = 1, f n = f n - 1 + f n - 2 , n 2 . Proposition. f n = φ n - (1 - φ ) n 5 , where φ = 1+ 5 2 1 . 618 ... is the golden ratio. February 15 Recursive structures, Page 3
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13.3 Rooted trees A rooted tree is a tree with a vertex designated as the root and is defined recursively as follows: 1. A single vertex v is a rooted tree, with the root v . 2. If T 1 , · · · , T n are disjoint rooted trees with roots r 1 , · · · , r n , then the graph T formed by starting with an additional vertex r and adding an edge between r to r i , i = 1 ...n , is also a rooted
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Unformatted text preview: tree with the root r . Each T i is called a subtree of T , and each r i is a child of r , and r is a parent of r i . A binary tree is a rooted tree of which each vertex has no more than 2 children. A full binary tree is a binary tree of which each vertex has either 0 or 2 children. The height h ( T ) of a rooted tree T is 1. 0 if T consists of a single vertex; February 15 Recursive structures, Page 4 13.3 Rooted trees 2. max { h ( T i ) : i = 1 ...n } + 1 if T 1 ...T n are subtrees of v . Fact. If T is a binary tree with n vertices, then n ≤ 2 h ( T )+1-1. February 15 Recursive structures, Page 5 13.4 Lower bound for sorting Theorem. Any sorting algorithm based on pair-wise comparisons must compare ≥ n log 2 n-cn times, for some constant c > 0 to sort n numbers. February 15 Recursive structures, Page 6...
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