1. Paul Tymann, Andrew Watkins,
Tree Traversal
Section 9.3
Longin Jan Latecki
Temple University
Based on slides by
Paul Tymann, Andrew Watkins,
and J. van Helden

2. Tree Anatomy
The children of a node are, themselves, trees, called subtrees.
Root
Level 0 R
Level 1 S T
Internal Node
Level 2 X U V W
Leaf
Level 3 Y Z
Child of X
Subtree
Parent of Z and Y

3. Tree Traversals
One of the most common operations performed on trees, are a tree traversals
A traversal starts at the root of the tree and visits every node in the tree exactly once
visit means to process the data in the node
Traversals are either depth-first or breadth-first

4. Breadth First Traversals
All the nodes in one level are visited
Followed by the nodes the at next level
Beginning at the root
For the sample tree
7, 6, 10, 4, 8, 13, 3, 5 7 6 10 4 8 13 3 5

5. Queue and stack
A queue is a sequence of elements such that each new element is added (enqueued) to one end, called the back of the queue, and an element is removed (dequeued) from the other end, called the front
A stack is a sequence of elements such that each new element is added (or pushed) onto one end, called the top, and an element is removed (popped) from the same end

6. Breadth first tree traversal with a queue
Enqueue root
While queue is not empty
Dequeue a vertex and write it to the output list
Enqueue its children left-to-right
Step Output Queue
0 a
1 a e,d
2 e d,i,b
3 d i,b,k,l
4 i b,k,l
5 b k,l,f
6 k l,f
7 l f
8 f g
9 g j,h
10 j h,m
11 h m
12 m c
13 c a b i f e d l g j h c m k 1 5 4 8 2 3 7 9 10 11 13 12 6

7. Depth-First Traversals
There are 6 different depth-first traversals
VLR (pre-order traversal)
VRL
LVR (in-order traversal)
RVL
RLV
LRV (post-order traversal)

8. Pre-order Traversal: VLR
Visit the node
Do a pre-order traversal of the left subtree
Finish with a pre-order traversal of the right subtree
For the sample tree
7, 6, 4, 3, 5, 10, 8, 13 7 6 10 4 8 13 3 5

9. Pre-order tree traversal with a stack
Push root onto the stack
While stack is not empty
Pop a vertex off stack, and write it to the output list
Push its children right-to-left onto stack
Step Output Stack
0 a
1 a d,e
2 e d,b,i
3 i d,b
4 b d,f
5 f d,g
6 g d,h,j
7 j d,h,m
8 m d,h,c
9 c d,h
10 h d
11 d l,k
12 k l
13 l a b i f e d l g j h c m k 1 4 3 5 2 11 13 6 7 10 9 8 12

10. Preorder Traversal Step 1: Visit r Step 2: Visit T1 in preorderTn
Step 2: Visit T1 in preorder
Step 3: Visit T2 in preorder
Step n+1: Visit Tn in preorder

11. Example A R E Y P M H J Q T M A J Y R H P Q T E

12. Ordering of the preorder traversal is the same a the Universal Address System with lexicographic ordering. A R E Y P M H J Q T 1 2 3
2.2
2.1
1.1 M A J Y R H P Q T E

13. In-order Traversal: LVR
Do an in-order traversal of the left subtree
Visit the node
Finish with an in-order traversal of the right subtree
For the sample tree
3, 4, 5, 6, 7, 8, 10, 13 7 6 10 4 8 13 3 5

14. Inorder Traversal Step 1: Visit T1 in inorder Step 2: Visit rTn
Step 2: Visit r
Step 3: Visit T2 in inorder
Step n+1: Visit Tn in inorder

15. Example A R E Y P M H J Q T J A M R Y P H Q T E

16. inorder (t)
if t != NIL: {
inorder (left[t]);
write (label[t]);
inorder (right[t]); }
Inorder Traversal on a binary search tree.

17. Post-order Traversal: LRV
Do a post-order traversal of the left subtree
Followed by a post-order traversal of the right subtree
Visit the node
For the sample tree
3, 5, 4, 6, 8, 13, 10, 7 7 6 10 4 8 13 3 5

18. Postorder Traversal Step 1: Visit T1 in postorderTn
Step 2: Visit T2 in postorder
Step n: Visit Tn in postorder
Step n+1: Visit r

19. Example A R E Y P M H J Q T J A R P Q T H Y E M

20. 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

21. Example
(x+y)2 + (x-3)/(y+2) + 2  / + x y – x 3 + y 2

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

23. Prefix Notation (Polish Notation)
Traverse in preorder (VLR) +  – / 2 x y 3 +  + x y 2 / – x 3 + y 2

24. 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

25. Example
+ / / –
+ / / –
+ / /
+ /
+ / 3

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

27. 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

28. Example
/ – / +
/ – / +
– / +
/ +
/ + 3