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Unformatted text preview: CSIS1118 oundations of Computer Science Foundations of Computer Science Recursion Recursion Hubert Chan ([O1,O3]; Chapters 4.3, 4.4) 1 athematical Induction Mathematical Induction Theorem: P(n) for all positive integers n. A proof by mathematical induction consists of two steps. asis step: (1) is true. Basis step: P(1) is true. Inductive step: for any positive integer k, if P(k) is true, then P(k+1) is true When we complete both steps, we have proved that P(n) is true for all positive integer n 1. Basic philosophy : [ P (1) k> 0 ( P ( k ) P ( k +1))] 2 n ≥ 1 P ( n ) ecursive Construction of Positive Integers Recursive Construction of Positive Integers Base Case : 1 is a positive integer Recursive Definition : If k is a positive integer, then (k+1) is also a positive integer. 3 ull Binary Trees ull Binary Trees hapter 4 3) hapter 4 3) Full Binary Trees Full Binary Trees (Chapter 4.3) (Chapter 4.3) Full Binary Trees (FBT) can be defined recursively Basis step : a single vertex (root) is a FBT Recursive step : If T 1 and T 2 are FBT’s, a root with 2 edges, ne connecting to the root of T nd the other to the root one connecting to the root of T 1 and the other to the root of T 2 , is also a FBT. Basis step Step 1 4 Step 2 ree Terminology ree Terminology Tree Terminology Tree Terminology Root: Top node (has no parent) root r Internal node: Node with children Leaf: Node with no children Internal ode x node y z leaf 5 Structural Induction Structural Induction Theorem. For any full binary tree T, int(T) = #. of internal nodes, leaf(T) = #. of leaves. Then, leaf(T) = int(T) + 1. 6 ecursive Algorithms ecursive Algorithms Recursive Algorithms Recursive Algorithms To solve P ( n ) for n ≥ 1, where P ( n ) is a problem of size n when n = 1, solve P (1) directly when n > 1, solve...
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 Spring '11
 Lin
 Mathematical Induction, Recursion, Natural number, positive integer, n1 n2, mathematical induction. induction

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