ICS 241

1
ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)
• Based on slides originally created by
• Dr. Michael P. Frank, Department of Computer
  Information Science Engineering at University
  of Florida

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Section 9.3 Tree Traversal

• Universal address systems
• Traversal algorithms
• Depth-first traversal
• Preorder traversal
• Inorder traversal
• Postorder traversal
• Infix/prefix/postfix notation

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Universal Address Systems

• A way to label the nodes of an ordered rooted
  tree
• Label the root with integer 0
• Label its children from left to right with 1, 2,
  3,
• For the next level down from vertex A, label its
  children with A.1, A.2, A.3,

4
Class Exercise

• Exercise 3. (p. 672)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

5
Traversal Algorithms

• Methods for visiting every node of an ordered
  rooted tree
• Most common traversals
• Preorder traversal
• Inorder traversal
• Postorder traversal

6
Preorder Traversal

• Steps
• Visit the root
• Visit the leftmost node in Preorder
• Visit the 2nd leftmost node in Preorder
• Visit the Nth node
• Example traversal
• 7, 3, 1, 0, 2, 5, 12, 9, 8, 11, 15

Example
7
3
12
1
5
9
15
0
2
8
11
7
Inorder Traversal

• Steps
• Visit the leftmost node in Inorder
• Visit the root
• Visit the 2nd leftmost node in Inorder
• Visit the Nth node
• Example traversal
• 0, 1, 2, 3, 5, 7, 8, 9, 11, 12, 15

Example
7
3
12
1
5
9
15
0
2
8
11
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Postorder Traversal

• Steps
• Visit the leftmost node in Postorder
• Visit the 2nd leftmost node in Postorder
• Visit the Nth node
• Visit the root
• Example traversal
• 0, 2, 1, 5, 3, 8, 11, 9, 15, 12, 7

Example
7
3
12
1
5
9
15
0
2
8
11
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Class Exercise

• Exercise 7. (p. 673)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

10
Infix/Prefix/Postfix Notation

• Can use an ordered, rooted tree to represent
  arithmetic expressions
• Leaves represent variables or numbers
• Interior nodes represent operations
• Operations are evaluated from the bottom of the
  tree to the top, and from the left node to the
  right node

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Infix/Prefix/Postfix Notation

• Infix
• ((16)-5)((2?5)/8)
• Prefix
• -165/?258
• Polish notation
• By Jan Lukasiewicz
• Infix
• 165-25?8/
• Reverse Polish notation

Example

-
/

5
?
8
1
6
2
5
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Class Exercise

• Exercise 23. (p. 673)
• Each pair of students should use only one sheet
  of paper while solving the class exercises