Section 9.3 by Andrew Watkins Tree Traversal Section 9.3 by Andrew Watkins CSE 2813 Discrete Structures

CSE 2813 Discrete Structures 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 Three common traversals are: Preorder traversal Inorder traversal Postorder traversal CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Traversal Let T be an ordered rooted tree with root r. Suppose T1, T2, …,Tn are the subtrees at r from left to right in T. r T1 T2 Tn CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Preorder Traversal Step 1: Visit r r T1 T2 Tn Step 2: Visit T1 in preorder Step 3: Visit T2 in preorder Step n+1: Visit Tn in preorder CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Example A R E Y P M H J Q T M A J Y R H P Q T E CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Inorder Traversal Step 1: Visit T1 in inorder r T1 T2 Tn Step 2: Visit r Step 3: Visit T2 in inorder Step n+1: Visit Tn in inorder CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Example A R E Y P M H J Q T J A M R Y P H Q T E CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Postorder Traversal Step 1: Visit T1 in postorder r T1 T2 Tn Step 2: Visit T2 in postorder Step n: Visit Tn in postorder Step n+1: Visit r CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Example A R E Y P M H J Q T J A R P Q T H Y E M CSE 2813 Discrete Structures

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 CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Example (x+y)2 + (x-3)/(y+2) + 2  / + x y – x 3 + y 2 CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Infix Notation Traverse in inorder adding parentheses for each operation +  – / 2 x y 3 ( ) ( ) ( ) x + y  2 + ( ) ( ) x – 3 / ( ) y + 2 CSE 2813 Discrete Structures

Prefix Notation (Polish Notation) Traverse in preorder +  – / 2 x y 3 +  + x y 2 / – x 3 + y 2 CSE 2813 Discrete Structures

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 CSE 2813 Discrete Structures

CSE 2813 Discrete Structures 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 CSE 2813 Discrete Structures

Postfix Notation (Reverse Polish) Traverse in postorder +  – / 2 x y 3 x y + 2  x 3 – y 2 + / + CSE 2813 Discrete Structures

Evaluating Postfix Notation 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 CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Example 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 + 3 CSE 2813 Discrete Structures

CSE 2813 Discrete Structures Exercises 7, 9, 10, 12, 13, 15, 16, 17, 18, 23, 24 CSE 2813 Discrete Structures