1. 4/17/2017
Section 9.3
Tree Traversal
ch9.3

2. Universal Address System
In ordered rooted trees, vertices may be labeled according to the following scheme:
choose a root node and label it 0
each of root’s k children are labeled, left to right, as 1, 2, … , k
for each vertex v at level n with label A, label its kv children left to right as A.1, A.2, … A.k

3. Universal Address System of an Ordered Rooted Tree

4. Traversal Algorithms
Traversal: procedure for visiting each vertex in an ordered tree for data access
Three most commonly used traversal algorithms are:
preorder
inorder
postorder

5. Preorder Traversal
In preorder traversal, the root vertex is visited first
Then the left subtree is visited using a preorder traversal
Then the right subtree is visited using a preorder traversal
Gives same ordering of vertices as the universal address system

6. Pre-order traversal in action
Original tree:
Results: K K I K I R W R K W O O O O D D

7. Inorder traversal
From the root vertex, proceed to the left subtree and perform an inorder traversal
Return to root and access the data there
Traverse the right subtree using inorder traversal

8. In-order traversal in action
Original tree:
Results: K K I I R W K R K W O O O O D D

9. Postorder traversal
From root node, proceed to left subtree and perform postorder traversal
Perform postorder traversal of right subtree
Access data at root vertex

10. Post-order traversal in actionK
Original tree:
Results: W K I I R O D O K W O R O D K

11. Infix, prefix and postfix notation
Ordered rooted trees (especially ordered binary trees) are useful in representing complicated expressions (e.g. compound propositions, arithmetic expressions)
A binary expression tree is a tree used to represent such an expression

12. Example 1
Create an ordered tree to represent the expression (x+y)2 + (x-4)/3
operands are represented as leaves
operators are represented as roots of subtrees

13. Example 1 Subtrees of binary expression tree for (x+y)2 + (x-4)/3:
4/17/2017
Example 1
Subtrees of binary expression tree for (x+y)2 + (x-4)/3:
Complete binary expression tree
for (x+y)2 + (x-4)/3: +
/ \
x y -
/ \ x +
/ \
^ 
/ \ / \
/ \ / \
x y x 4 ^
/ \
x y 
/ \ x
ch9.3

14. Traversing binary expression tree
Inorder traversal of binary expression tree produces original expression (without parentheses), in infix order
Preorder traversal produces a prefix expression
Postorder traversal produces a postfix expression

15. Prefix expressions The prefix version of the expression
(x+y)2 + (x-4)/3 is:
+ ^ + x y 2 / - x 4 3
Evaluating prefix expressions:
Read expression right to left
When an operator is encountered, apply it to the previous operand (if unary) or operands (if binary) , placing the result back into the expression where the subexpression had been

16. Example 2 + * / 4 2 3 9 // original expression
+ * // 4/2 evaluated
// 2*3 evaluated
15 // 6+9 evaluated

17. Example 3 * - + 4 3 5 / + 2 4 3 // original expression
* / 6 3 // 2+4 evaluated
* // 6/3 evaluated
* // 4+3 evaluated
* // 7-5 evaluated
4 // 2*2 evaluated

18. Postfix expressions Also known as reverse Polish expressions
Like infix, they are evaluated left to right
Like prefix, they are unambiguous, not requiring parentheses
To evaluate:
read expression left to right; as soon as an operator is encountered, perform the operation and place the result back in the expression

19. Postfix expressions Simple expression: Original Expression: A + B
Postfix Equivalent: A B +
Compound expression with parentheses:
original: (A + B) * (C - D)
postfix: A B + C D - *
Compound expression without parentheses:
original: A + B * C - D
postfix: A B C * + D -

20. Example 4 6 3 / 4 2 * + // original expression
2 4 2 * + // 6/3 evaluated
// 4*2 evaluated
10 // 2+8 evaluated

21. Example 5 5 4 * 10 2 - 2 / + 3 * // original expression
/ + 3 * // 5*4 evaluated
/ + 3 * // 10-2 evaluated
* // 8/2 evaluated
24 3 * // 20+4 evaluated
72 // 24*3 evaluated

22. Using rooted trees to represent compound propositions
Works exactly the same way as arithmetic expressions
Innermost expression is bottom left subtree, with proposition(s) as leaf(s) and operator as root
Root vertex is operator of outermost expression
Using various traversal methods, can produce infix, prefix and postfix versions of compound proposition

23. Example 6 Subtrees: Complete binary expression tree:
Find the ordered rooted tree representing the compound proposition ((p  q)  (p  q)
Subtrees:
Complete binary
expression tree: 
/ \
p q  | p  | q 
/ \
 
| / \
  
/ \ | |
p q p q 
/ \
 
| |
p q  | 
/ \
p q

24. Example 6 Preorder traversal yields the expression:
/ \
 
| / \
  
/ \ | |
p q p q
Preorder traversal yields the expression:
   p q   p  q
Postorder traversal yields the expression:
p q   p  q   

25. Section 9.3
Tree Traversal