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Chapter 10, Section 10.3 Tree Traversal These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published by McGraw Hill, Boston, MA, 2006. They are intended for classroom use only and are not a substitute for reading the textbook.

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Universal Address Systems To totally order the vertices of on orered rooted tree: 1.Label the root with the integer 0. 2.Label its k children (at level 1) from left to right with 1, 2, 3, …, k. 3.For each vertex v at level n with label A, label its k v children, from left to right, with A.1, A.2, A.3, …, A.k v. This labeling is called the universal address system of the ordered rooted tree.

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Universal Address Systems

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Traversal Algorithms A traversal algorithm is a procedure for systematically visiting every vertex of an ordered rooted tree –An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered The three common traversals are: –Preorder traversal –Inorder traversal –Postorder traversal

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Traversal Let T be an ordered rooted tree with root r. Suppose T 1, T 2, …,T n are the subtrees at r from left to right in T. r T1T1 T2T2 TnTn

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Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder

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Preorder Traversal

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Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence:

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The Preorder Traversal of T In which order does a preorder traversal visit the vertices in the ordered rooted tree T shown to the left? Preorder: Visit root, then visit subtrees left to right.

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The Preorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved Preorder: Visit root, then visit subtrees left to right.

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The Preorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Preorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Preorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn

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Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence:

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The Inorder Traversal of T In which order does an inorder traversal visit the vertices in the ordered rooted tree T shown to the left? Inorder: Visit leftmost tree, visit root, visit other subtrees left to right.

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The Inorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved Inorder: Visit leftmost tree, visit root, visit other subtrees left to right.

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The Inorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Inorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Inorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn

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Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence:

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The Postorder Traversal of T In which order does a postorder traversal visit the vertices in the ordered rooted tree T shown to the left? Postorder: Visit subtrees left to right, then visit root.

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The Postorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved Postorder: Visit subtrees left to right, then visit root.

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The Postorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Postorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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The Postorder Traversal of T © The McGraw-Hill Companies, Inc. all rights reserved

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Representing Arithmetic Expressions Complicated arithmetic expressions can be represented by an ordered rooted tree –Internal vertices represent operators –Leaves represent operands Build the tree bottom-up –Construct smaller subtrees –Incorporate the smaller subtrees as part of larger subtrees

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Example (x+y) 2 + (x-3)/(y+2) + x y 2 – x 3 + y 2 / +

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Infix Notation + – + / + 2 x y x 3 y 2 Traverse in inorder adding parentheses for each operation x + y () 2 () + x – 3() / y + 2() () ()

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Prefix Notation (Polish Notation) Traverse in preorder: x + y 2 + x – 3 / y + 2 + – + / + 2 x y x 3 y 2

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Evaluating Prefix Notation In an prefix expression, a binary operator precedes its two operands The expression is evaluated right-left Look for the first operator from the right Evaluate the operator with the two operands immediately to its right

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Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 + 2 1 3

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Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 + x – 3 / y + 2 + – + / + 2 x y x 3 y 2

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In an postfix expression, a binary operator follows its two operands The expression is evaluated left-right Look for the first operator from the left Evaluate the operator with the two operands immediately to its left Evaluating Postfix Notation

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Example 3 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 2 1 +

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Homework Exercises Do questions : 7, 9, 10, 12, 13, 15, 16, 17, 18, 23, 24

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Conclusion In this chapter we have studied: –Trees –Applications of Trees –Tree Traversal

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