Program to convert Infix notation to Expression Tree

Given a string representing infix notation. The task is to convert it to an expression tree.
Expression Tree is a binary tree where the operands are represented by leaf nodes and operators are represented by intermediate nodes. No node can have a single child.

Construction of Expression tree

The algorithm follows a combination of shunting yard along with postfix-to-expression tree conversion.

Consider the below line:

((s[i]!='^' && p[stC.top()]>=p[s[i]]) || 
(s[i]=='^' && p[stC.top()]>p[s[i]]))) 

You might remember that unlike ‘+’, ‘-‘, ‘*’ and ‘/’; ‘^’ is right associative. 
In simpler terms, a^b^c is a^(b^c) not (a^b)^c. So it must be evaluated from the right.



Now lets take a look at how does the algorithm work,(Take a quick glance at the code to get a better idea of the variables used) 

Let us have an expression s = ((a+b)*c-e*f)
currently both the stacks are empty- 
(we'll use C to denote the char stack and N for node stack)
    s[0] = '('            ((a+b)*c-e*f)
                          ^
        C|(|, N| |
    
    s[1] = '('           ((a+b)*c-e*f)
                          ^
         |(|  
        C|(|, N| |
                         
    s[2] = 'a'            ((a+b)*c-e*f)
                            ^
         |(|  
        C|(|, N|a|
                         
    s[3] = '+'            ((a+b)*c-e*f)
                             ^
         |+|
         |(|  
        C|(|, N|a|
        
    s[4] = 'b'             ((a+b)*c-e*f)
                               ^
         |+|
         |(|   |b|
        C|(|, N|a|
         
    s[5] = ')'             ((a+b)*c-e*f)
                                ^
         |+|            t = '+'         +
         |(|   |b|  ->  t1= 'b'        / \    ->  
        C|(|, N|a|      t2= 'a'       a   b      C|(|, N|+|
        
    s[6] = '*'              ((a+b)*c-e*f)
                                  ^
         |*|  
        C|(|, N|+|
    
    s[7] = 'c'              ((a+b)*c-e*f)
                                   ^
         |*|   |c|
        C|(|, N|+|
        
    s[8] = '-'   ((a+b)*c-e*f)              now (C.top(*)>s[8](-))
                         ^    t = '*'        *
         |*|   |c|            t1 = c        / \  ->    |-|
        C|(|, N|+|            t2 = +       +   c      C|(|, N|*|
                                          / \
                                         a   b
    s[9] = 'e'            ((a+b)*c-e*f)
                                   ^
         |-|   |e|
        C|(|, N|*|
        
    s[10] = '*'            ((a+b)*c-e*f)      now (C.top(-)>s[10](*))
                                     ^
         |*|                          
         |-|   |e| 
        C|(|, N|*|
    
    s[11] = 'f'             ((a+b)*c-e*f)
                                       ^
         |*|   |f|
         |-|   |e|
        C|(|, N|*|
    
    s[12] = ')'             ((a+b)*c-e*f)
       1>                               ^
         |*|   |f|         t = '*'          *        
         |-|   |e|  ->     t1= 'f'  ->     / \  ->   |-|   |*|
        C|(|, N|*|         t2= 'e'        e   f     C|(|, N|*|
        
       2>                               
                           t = '-'           -        
         |-|   |*|  ->     t1= '*'  ->     /   \  ->   
        C|(|, N|*|         t2= '*'        *     *     C| |, N|-|
                                         / \   / \
                                        +   c  e  f
                                       / \
                                      a   b 
      now make (-) the root of the tree

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Tree Structure
typedef struct node 
{
    char data;
    struct node *left, *right;
} * nptr;
  
// Function to create new node
nptr newNode(char c)
{
    nptr n = new node;
    n->data = c;
    n->left = n->right = nullptr;
    return n;
}
  
// Function to build Expression Tree
nptr build(string& s)
{
  
    // Stack to hold nodes
    stack<nptr> stN;
  
    // Stack to hold chars
    stack<char> stC;
    nptr t, t1, t2;
  
    // Prioritising the operators
    int p[123] = { 0 };
    p['+'] = p['-'] = 1, p['/'] = p['*'] = 2, p['^'] = 3,
    p[')'] = 0;
  
    for (int i = 0; i < s.length(); i++) 
    {
        if (s[i] == '(') {
  
            // Push '(' in char stack
            stC.push(s[i]);
        }
  
        // Push the operands in node stack
        else if (isalpha(s[i])) 
        {
            t = newNode(s[i]);
            stN.push(t);
        }
        else if (p[s[i]] > 0) 
        {
            // If an operator with lower or
            // same associativity appears
            while (
                !stC.empty() && stC.top() != '('
                && ((s[i] != '^' && p[stC.top()] >= p[s[i]])
                    || (s[i] == '^'
                        && p[stC.top()] > p[s[i]]))) 
            {
  
                // Get and remove the top element
                // from the character stack
                t = newNode(stC.top());
                stC.pop();
  
                // Get and remove the top element
                // from the node stack
                t1 = stN.top();
                stN.pop();
  
                // Get and remove the currently top
                // element from the node stack
                t2 = stN.top();
                stN.pop();
  
                // Update the tree
                t->left = t2;
                t->right = t1;
  
                // Push the node to the node stack
                stN.push(t);
            }
  
            // Push s[i] to char stack
            stC.push(s[i]);
        }
        else if (s[i] == ')') {
            while (!stC.empty() && stC.top() != '('
            {
                t = newNode(stC.top());
                stC.pop();
                t1 = stN.top();
                stN.pop();
                t2 = stN.top();
                stN.pop();
                t->left = t2;
                t->right = t1;
                stN.push(t);
            }
            stC.pop();
        }
    }
    t = stN.top();
    return t;
}
  
// Function to print the post order
// traversal of the tree
void postorder(nptr root)
{
    if (root) 
    {
        postorder(root->left);
        postorder(root->right);
        cout << root->data;
    }
}
  
// Driver code
int main()
{
    string s = "(a^b^(c/d/e-f)^(x*y-m*n))";
    s = "(" + s;
    s += ")";
    nptr root = build(s);
    
    // Function call
    postorder(root);
  
    return 0;
}
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Output
abcd/e/f-xy*mn*-^^^

The time Complexity is O(n) as each character is accessed only once. 
The space Complexity is O(n) as (char_stack + node_stack) <= n
 

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