# 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,

```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 = '('            ((a+b)*c-e*f)
^
C|(|, N| |

s = '('           ((a+b)*c-e*f)
^
|(|
C|(|, N| |

s = 'a'            ((a+b)*c-e*f)
^
|(|
C|(|, N|a|

s = '+'            ((a+b)*c-e*f)
^
|+|
|(|
C|(|, N|a|

s = 'b'             ((a+b)*c-e*f)
^
|+|
|(|   |b|
C|(|, N|a|

s = ')'             ((a+b)*c-e*f)
^
|+|            t = '+'         +
|(|   |b|  ->  t1= 'b'        / \    ->
C|(|, N|a|      t2= 'a'       a   b      C|(|, N|+|

s = '*'              ((a+b)*c-e*f)
^
|*|
C|(|, N|+|

s = 'c'              ((a+b)*c-e*f)
^
|*|   |c|
C|(|, N|+|

s = '-'   ((a+b)*c-e*f)              now (C.top(*)>(-))
^    t = '*'        *
|*|   |c|            t1 = c        / \  ->    |-|
C|(|, N|+|            t2 = +       +   c      C|(|, N|*|
/ \
a   b
s = 'e'            ((a+b)*c-e*f)
^
|-|   |e|
C|(|, N|*|

s = '*'            ((a+b)*c-e*f)      now (C.top(-)>(*))
^
|*|
|-|   |e|
C|(|, N|*|

s = 'f'             ((a+b)*c-e*f)
^
|*|   |f|
|-|   |e|
C|(|, N|*|

s = ')'             ((a+b)*c-e*f)
1>                               ^
|*|   |f|         t = '*'          *
|-|   |e|  ->     t1= 'f'  ->     / \  ->   |-|   |*|
C|(|, N|*|         t2= 'e'        e   f     C|(|, N|*|

2>
t = '-'           -
|-|   |*|  ->     t1= 'f'  ->     /   \  ->
C|(|, N|*|         t2= 'e'        *     *     C| |, N|-|
/ \   / \
+   c  e  f
/ \
a   b
now make (-) the root of the tree
```

Below is the implementation of the above approach:

 `// C++ implementation of the approach ` `#include ` `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 stN; ` ` `  `    ``// Stack to hold chars ` `    ``stack<``char``> stC; ` `    ``nptr t, t1, t2; ` ` `  `    ``// Prioritising the operators ` `    ``int` `p = { 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); ` `    ``postorder(root); ` ` `  `    ``return` `0; ` `} `

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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