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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

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

12. 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

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

14. 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

15. CSE 2813 Discrete Structures
Example
+ / / –
+ / / –
+ / /
+ /
+ / 3
CSE 2813 Discrete Structures

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

17. 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

18. CSE 2813 Discrete Structures
Example
/ – / +
/ – / +
– / +
/ +
/ + 3
CSE 2813 Discrete Structures

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