1. CS 201 Computer Systems Programming Chapter 7 “Printing Binary Trees”
Herbert G. Mayer, PSU CS
Status 11/1/2013

2. Syllabus Arithmetic Expressions and Trees Infix Without Parentheses
Infix With Parentheses
Postfix Without Parentheses
Prefix Without Parentheses
Interesting Examples
Use of Postfix

3. Arithmetic Expressions and Trees
Three typical notations for dyadic operations:
Infix notation: write as the first the left operand, reading left-to- right, then list the dyadic operator, finally list the right operand
For CPU: Order will not work for code emission, as the CPU needs both operands for processing the operator
For humans: requires parentheses for proper operator precedence
Note exception: programming language APL
Postfix notation: write left operand first, then list the right operand, finally the operator
This order will work for code emission, as operator has both operands available at processing time
Needs no parentheses, and still obeys operator precedence
Postfix notation AKA Polish Postfix, after Jan Łukasiewicz, 1920
Prefix notation: First list the operator, next the first (left) operand, finally the second (right) operand

4. Arithmetic Expressions and Trees
a + x ^ c
Infix: ( a + ( x ^ c ) )
Postfix: a x c ^ +
Prefix: + a ^ x c + a ^ x c
^ stands for exponentiation operator, with highest precedence: higher than * or /

5. Arithmetic Expressions and Trees
( x – a ) / b
Infix: ( ( x – a ) ) / b
Postfix: x a – b /
Prefix: / – x a b / - b x a
/ stands for division operator, with higher precedence than, say, –

6. Arithmetic Expressions and Trees
a ^ ( b – c ) / d
Infix: ( ( a ^ ( b – c ) ) / d )
Postfix: a b c - ^ d /
Prefix: / ^ a – b c d / ^ d a - b c

7. Data Structure to Print Trees
// node has class: literal, identifier, or operator.
// Parenthesized expressions have been reduced: no ( )
typedef enum { Literal, Identifier, Operator } NodeClass;
typedef struct NodeType * NodePtr; // forward
// actual node structure; using the forward pointers
typedef struct NodeType {
NodeClass Class; // 3 classes. Not C++ ‘class’
char Symbol; // stores ident or small literal
int LitVal; // if Class == Literal: its value
NodePtr Left; // left subtree
NodePtr Right; // right subtree
} s_node_tp;

8. Infix Without Parentheses
// Print in infix notation without parentheses ( )
void Print_No_Paren( NodePtr Root )
{ // Print_No_Paren
if ( Root ) {
Print_No_Paren ( Root->Left );
if ( Root->Class == Literal ) {
printf( "%d", Root->LitVal );
}else{
printf( "%c", Root->Symbol );
} //end if
Print_No_Paren ( Root->Right );
} //end Print_No_Paren
Input: ( a + x ) / b prints as: a + x / b  misleading

9. Infix With Parentheses
// Print in infix notation with parentheses ( and )
// though prints too many ( ) pairs
void Print_Infix( NodePtr Root )
{ // Print_Infix
if ( Root ) {
if ( Root->Class == Operator ) { // should have inverted
printf( "(" );
} //end if
Print_Infix( Root->Left );
if ( Root->Class == Literal ) {
printf( "%d", Root->LitVal );
}else{
printf( "%c", Root->Symbol );
Print_Infix( Root->Right );
if ( Root->Class == Operator ) {
printf( ")" );
} //end Print_Infix
Input: ( a + x ) / b prints as: ( ( a + x ) / b ) -- OK

10. Postfix Without Parentheses
// Print in Polish Postfix notation, no parentheses
void Print_Postfix( NodePtr Root )
{ // Print_Postfix
if ( Root ) {
Print_Postfix( Root->Left );
Print_Postfix( Root->Right );
if ( Root->Class == Literal ) {
printf( "%d", Root->LitVal );
}else{
printf( "%c", Root->Symbol );
} //end if
} //end Print_Postfix
Input: a ^ ( b – c ) / d prints as: a b c - ^ d / -- OK

11. Prefix Without Parentheses
// Prefix: operator executes when 2 operands found
void Print_Prefix( NodePtr Root )
{ // Print_Prefix
if ( Root ) {
if ( Root->Class == Literal ) {
printf( "%d", Root->LitVal );
}else{
printf( "%c", Root->Symbol );
} //end if
Print_Prefix( Root->Left );
Print_Prefix( Root->Right );
} //end Print_Prefix
Input: ( a + x ) / b prints as: / + a x b -- OK

12. Interesting Examples Input 1: a + b * c ^ ( x – 2 * d ) / ( e – f )
Infix: ( a + ( ( b * ( c ^ ( x – ( 2 * d ) ) ) ) / ( e – f ) ) )
Postfix: a b c x 2 d * - ^ * e f - / +
Prefix: + a / * b ^ c – x * 2 d – e f
Input 2: 4 / x ^ ( k – l / m ) * 8 * x - & 9 + n
Infix: ( ( ( ( ( 4 / ( x ^ ( k - ( l / m ) ) ) ) * 8 ) * x ) - ( & 9 ) ) + n )
Postfix: 4 x k l m / - ^ / 8 * x * 9 & - n +
Prefix: + - * * / 4 ^ x – k / l m 8 x & 9 n

13. Use of Postfix
Postfix, AKA Polish Postfix notation is a natural for code generation, not just for stack machines
Operands are needed first: Two for dyadic, or one for monadic operations
Once generated and available on stack, stack machine can execute the next operation
Easy for compiler writer, natural for stack machine
Stack poor for execution, as all references are through memory: top of stack
Even a GPR architecture needs both operands available somewhere (in regs) to execute operator

14. References Łukasiewicz: http://www.calculator.org/Lukasiewicz.aspx