The document discusses binary expression trees, which are a type of binary tree used to represent arithmetic expressions. A binary expression tree has operands as leaf nodes and operators as internal nodes. The left and right subtrees of each operator node represent subexpressions that must be evaluated before applying the operator. The document provides examples of binary expression trees and discusses traversing them using inorder, preorder, and postorder traversals. Algorithms are presented for building a binary expression tree from a prefix notation expression and for evaluating an expression by scanning it and using operator and value stacks.

1. Data Structures
Presentation By:
Amna Iqbal
Amna Muzzafar
Asma Iqbal
Faiza Zahid
Maryam Tariq
Sumaira
Shabana kausar
Shamsa
Tahseen Fatima
Zeerak

2. 2
An application of binary trees
Binary Expression Trees

3. 3
A special kind of binary tree in which:
• The leaves of a binary expression tree are operands, such as
constants or variable names , and the other nodes contain
operators
• Each leaf node contains a single operand
• Each nonleaf node contains a single binary operator
• The left and right subtrees of an operator node represent
subexpressions that must be evaluated before applying the
operator at the root of the subtree.
A Binary Expression Tree is . . .

4. 4
A Binary Expression Tree
What value does it have?
( 4 + 2 ) * 3 = 18
‘*’
‘+’
‘4’
‘3’
‘2’

5. 5
A Four-Level Binary Expression
‘*’
‘-’
‘8’ ‘5’
‘/’
‘+’
‘4’
‘3’
‘2’

6. Expression Tree Examples
Inorder Traversal ResultExpression TreeExpression
a + 3(a+3)
3+4*5-9+63+(4*5-(9+6))
log xlog(x)
n !n!
+
3a
+
-3
*
54
+
69
log
x
!
n

7. Infix Expressions
• When you write an arithmetic expression such as
B * C, the form of the expression provides you
with information so that you can interpret it
correctly. In this case we know that the variable B
is being multiplied by the variable C since the
multiplication operator * appears between them
in the expression This type of notation is referred
to as infix since the operator is in between the two
operands that it is working on.
7

8. Prefix Expressions
• Prefix expression notation requires that all
operators precede the two operands that
they work on.
• A + B * C would be written as +A*BC in
prefix. The multiplication operator comes
immediately before the operands B and C,
denoting that * has precedence over +. The
addition operator then appears before the A
and the result of the multiplication.
8

9. Postfix Expressions
• Postfix requires that its operators come after
the corresponding operands.
• A + B * C would be written as ABC*+ in postfix.
The multiplication operator appears
immediately after the B and the C, denoting
that * has precedence, with + coming after.
9

10. Examples of Infix, Prefix, and
Postfix
10
Infix Expression Prefix Expression Postfix Expression
A + B + A B A B +
A + B * C + A * B C A B C * +

11. 11
Easy to generate the infix, prefix, postfix
expressions (how?)
Infix: ( ( 8 - 5 ) * ( ( 4 + 2 ) / 3 ) )
Prefix: * - 8 5 / + 4 2 3
Postfix: 8 5 - 4 2 + 3 / *
‘*’
‘-’
‘8’ ‘5’
‘/’
‘+’
‘4’
‘3’
‘2’

12. Infix to Postfix conversion (manual)
• An Infix to Postfix manual conversion algorithm is:
1. Completely parenthesize the infix expression according to order of priority
you want.
2. Move each operator to its corresponding right parenthesis.
3. Remove all parentheses.
• Examples:
3 + 4 * 5 (3 + (4 * 5) ) 3 4 5 * +
a / b ^ c – d * e – a * c ^ 3 ^ 4 a b c ^ / d e * a c 3 4 ^ ^ * - -
((a / (b ^ c)) – ((d * e) – (a * (c ^ (3 ^ 4) ) ) ) )
Using normal mathematical
operator precedence
Not using normal mathematical
operator precedence

13. 13
Levels Indicate Precedence
The levels of the nodes in the tree indicate their
relative precedence of evaluation (we do not need
parentheses to indicate precedence).
Each operator has a precedence level. Operators of
higher precedence are used before operators of lower
precedence
Operations at higher levels of the tree are
evaluated later than those below them.
The operation at the root is always the last
operation performed.

14. Tree Traversal
• A tree traversal is a specific order in
which to trace the nodes of a tree.
14

15. Types Of Tree Traversal
• There are 3 common types of tree
traversals.
• In-order.
• 2. Pre-order.
• 3. Post-order.
15

16. 16
Inorder Traversal:
In-order: left, root, right (A + H) / (M - Y)
‘/’
‘+’
‘A’ ‘H’
‘-’
‘M’ ‘Y’
tree
Print left subtree first Print right subtree last
Print second

17. 17
Preorder Traversal:
Pre-order: root, left, right / + A H - M Y
‘/’
‘+’
‘A’ ‘H’
‘-’
‘M’ ‘Y’
tree
Print left subtree second Print right subtree last
Print first

18. 18
‘/’
‘+’
‘A’ ‘H’
‘-’
‘M’ ‘Y’
tree
Print left subtree first Print right subtree second
Print last
Postorder Traversal:
Post-order: left, right, root A H + M Y - /

19. 19
Building a Binary Expression Tree from an
expression in prefix notation
• Insert new nodes, each time moving to the left until
an operand has been inserted.
• Backtrack to the last operator, and put the next node
to its right.
• Continue in the same pattern.

20. Algorithm for Expression tree

21. Algorithm for Expression tree
1 Scan expression from left to right until end of string
2 If (symbol = operand)
P= new node
P.data= symbol
Push(P)
Else
Ptr1= Pop()
Ptr2= Pop()
P= new node
P.data= symbol
P.Lchild= Ptr2
P.Rchild=Ptr1
Push(P)
3 Exit

22. How to scan an expression
1. While there are still tokens to be read in,
1.1 Get the next token.
1.2 If the token is:
1.2.1 A number: push it onto the value stack.
1.2.2 A variable: get its value, and push onto the value stack.
1.2.3 A left parenthesis: push it onto the operator stack.
1.2.4 A right parenthesis:
1 While the thing on top of the operator stack is not a
left parenthesis,
1 Pop the operator from the operator stack.
2 Pop the value stack twice, getting two operands.
3 Apply the operator to the operands, in the correct order.
4 Push the result onto the value stack.
2 Pop the left parenthesis from the operator stack, and discard it.

23. How to scan an expression
1.2.5 An operator (call it thisOp):
1 While the operator stack is not empty, and the top
thing on the
operator stack has the same or greater precedence as
thisOp,
1 Pop the operator from the operator stack.
2 Pop the value stack twice, getting two operands.
3 Apply the operator to the operands, in the correct
order.
4 Push the result onto the value stack.
2 Push thisOp onto the operator stack.

24. How to scan an expression
2. While the operator stack is not empty,
1 Pop the operator from the operator stack.
2 Pop the value stack twice, getting two operands.
3 Apply the operator to the operands, in the correct
order.
4 Push the result onto the value stack.
3. At this point the operator stack should be empty,
and the value
stack should have only one value in it, which is the
final result