Prefix, Postfix, Infix Notation. Infix Notation  To add A, B, we write A+B  To multiply A, B, we write AB  The operators ('+' and '') go in between.

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Presentation transcript:

Prefix, Postfix, Infix Notation

Infix Notation  To add A, B, we write A+B  To multiply A, B, we write A*B  The operators ('+' and '*') go in between the operands ('A' and 'B')  This is "Infix" notation.

Prefix Notation  Instead of saying "A plus B", we could say "add A,B " and write + A B  "Multiply A,B" would be written * A B  This is Prefix notation.

Postfix Notation  Another alternative is to put the operators after the operands as in A B + and A B *  This is Postfix notation.

 The terms infix, prefix, and postfix tell us whether the operators go between, before, or after the operands. Pre A In B Post

Parentheses  Evaluate 2+3*5.  + First: (2+3)*5 = 5*5 = 25  * First: 2+(3*5) = 2+15 = 17  Infix notation requires Parentheses.

What about Prefix Notation?  + 2 * 3 5 = = + 2 * 3 5 = = 17  * = = * = * 5 5 = 25  No parentheses needed!

Postfix Notation  * + = = * + = = 17  * = = * = 5 5 * = 25  No parentheses needed here either!

Conclusion:  Infix is the only notation that requires parentheses in order to change the order in which the operations are done.

Fully Parenthesized Expression  A FPE has exactly one set of Parentheses enclosing each operator and its operands.  Which is fully parenthesized? ( A + B ) * C ( ( A + B) * C ) ( ( A + B) * ( C ) )

Infix to Prefix Conversion Move each operator to the left of its operands & remove the parentheses: ( ( A + B) * ( C + D ) )

Infix to Prefix Conversion Move each operator to the left of its operands & remove the parentheses: ( + A B * ( C + D ) )

Infix to Prefix Conversion Move each operator to the left of its operands & remove the parentheses: * + A B ( C + D )

Infix to Prefix Conversion Move each operator to the left of its operands & remove the parentheses: * + A B + C D Order of operands does not change!

Infix to Postfix ( ( ( A + B ) * C ) - ( ( D + E ) / F ) ) A B + C * D E + F / -  Operand order does not change!  Operators are in order of evaluation!

Computer Algorithm FPE Infix To Postfix  Assumptions: 1. Space delimited list of tokens represents a FPE infix expression 2. Operands are single characters. 3. Operators +,-,*,/

FPE Infix To Postfix  Initialize a Stack for operators, output list  Split the input into a list of tokens.  for each token (left to right): if it is operand: append to output if it is '(': push onto Stack if it is ')': pop & append till '('

FPE Infix to Postfix ( ( ( A + B ) * ( C - E ) ) / ( F + G ) )  stack:  output: []

FPE Infix to Postfix ( ( A + B ) * ( C - E ) ) / ( F + G ) )  stack: (  output: []

FPE Infix to Postfix ( A + B ) * ( C - E ) ) / ( F + G ) )  stack: ( (  output: []

FPE Infix to Postfix A + B ) * ( C - E ) ) / ( F + G ) )  stack: ( ( (  output: []

FPE Infix to Postfix + B ) * ( C - E ) ) / ( F + G ) )  stack: ( ( (  output: [A]

FPE Infix to Postfix B ) * ( C - E ) ) / ( F + G ) )  stack: ( ( ( +  output: [A]

FPE Infix to Postfix ) * ( C - E ) ) / ( F + G ) )  stack: ( ( ( +  output: [A B]

FPE Infix to Postfix * ( C - E ) ) / ( F + G ) )  stack: ( (  output: [A B + ]

FPE Infix to Postfix ( C - E ) ) / ( F + G ) )  stack: ( ( *  output: [A B + ]

FPE Infix to Postfix C - E ) ) / ( F + G ) )  stack: ( ( * (  output: [A B + ]

FPE Infix to Postfix - E ) ) / ( F + G ) )  stack: ( ( * (  output: [A B + C ]

FPE Infix to Postfix E ) ) / ( F + G ) )  stack: ( ( * ( -  output: [A B + C ]

FPE Infix to Postfix ) ) / ( F + G ) )  stack: ( ( * ( -  output: [A B + C E ]

FPE Infix to Postfix ) / ( F + G ) )  stack: ( ( *  output: [A B + C E - ]

FPE Infix to Postfix / ( F + G ) )  stack: (  output: [A B + C E - * ]

FPE Infix to Postfix ( F + G ) )  stack: ( /  output: [A B + C E - * ]

FPE Infix to Postfix F + G ) )  stack: ( / (  output: [A B + C E - * ]

FPE Infix to Postfix + G ) )  stack: ( / (  output: [A B + C E - * F ]

FPE Infix to Postfix G ) )  stack: ( / ( +  output: [A B + C E - * F ]

FPE Infix to Postfix )  stack: ( / ( +  output: [A B + C E - * F G ]

FPE Infix to Postfix )  stack: ( /  output: [A B + C E - * F G + ]

FPE Infix to Postfix  stack:  output: [A B + C E - * F G + / ]

Problem with FPE  Too many parentheses.  Establish precedence rules: My Dear Aunt Sally  We can alter the previous program to use the precedence rules.

Infix to Postfix  Initialize a Stack for operators, output list  Split the input into a list of tokens.  for each token (left to right): if it is operand: append to output if it is '(': push onto Stack if it is ')': pop & append till '(' if it in '+-*/': while peek has precedence ≥ it: pop & append push onto Stack pop and append the rest of the Stack.