"3 * 4 + 5"

"x + (4 - 5 * y"   // not grammatically correct

"x + )y - z("      // not grammatically correct

"x + * y"          // not grammatically correct


A "parser" is an algorithm that should convert the
strings of grammatically correct expressions into
a tree data structure that the interpreter/compiler
can use.


"3 + 4 * 5"
              +         abstract syntax tree
             / \        expression tree
            3   *
               / \
              4   5

"3 * 4 + 5"
              +
             / \
            *   5
           / \
          3   4

"(3 + 4) * 5"
              *
             / \
            +   5
           / \
          3   4


"if (x == y) print(x) else print(y)"

         if           abstract syntax tree
        / | \
       /  |  \
      /   |   \
   ==   print  print
  / \     |      |
 x   y    x      y