Tree Traversal

1
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
R
Level 0
Level 1
S
T
Internal Node
X
U
V
W
Level 2
Leaf
Y
Z
Level 3
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

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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

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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
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Depth-First Traversals

• There are 8 different depth-first traversals
• VLR (pre-order traversal)
• VRL
• LVR (in-order traversal)
• RVL
• RLV
• LRV (post-order traversal)

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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
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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
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Preorder Traversal
Step 1 Visit r
Step 2 Visit T1 in preorder
Step 3 Visit T2 in preorder
Step n1 Visit Tn in preorder
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Example
A
R
E
Y
P
M
H
J
Q
T
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Ordering of the preorder traversal is the same a
the Universal Address System with lexicographic
ordering.
0
1
2
3
2.2
2.1
1.1
2.2.1 2.2.2 2.2.3
A
R
E
Y
P
M
H
J
Q
T
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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
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Inorder Traversal
Step 1 Visit T1 in inorder
Step 2 Visit r
Step 3 Visit T2 in inorder
Step n1 Visit Tn in inorder
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Example
A
R
E
Y
P
M
H
J
Q
T
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inorder (t) if t ! NIL inorder
(leftt) write (labelt) inorder (rightt)

Inorder Traversal on a binary search tree.
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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
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Postorder Traversal
Step 1 Visit T1 in postorder
Step 2 Visit T2 in postorder
Step n Visit Tn in postorder
Step n1 Visit r
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Example
A
R
E
Y
P
M
H
J
Q
T
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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

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Example

• (xy)2 (x-3)/(y2)

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Infix Notation

• Traverse in inorder (LVR) adding parentheses for
  each operation

x
y
2


x

3
y

2
?
/
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Prefix Notation(Polish Notation)

• Traverse in preorder (VLR)

x
y
2


x

3
y

2
?
/
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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

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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
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Postfix Notation(Reverse Polish)

• Traverse in postorder (LRV)

x
y
2


x

3
y

2
?
/
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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

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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