An expression's parentheses, braces, and brackets are balanced when every opener has a closer of the same type to its right, every closer has an opener of the same type to its left, and no delimiter closes while a delimiter opened after it remains open — properties that counting occurrences of each type cannot verify, since equal counts are necessary but not sufficient. The decisive structural property is that during a left-to-right scan, any closer must correspond to the most recent unclosed opener, which is exactly the last-in-first-out discipline of a stack: openers are pushed, each closer must match the type at the top and trigger a pop, and the expression is balanced only if no mismatch or underflow occurs and the stack is empty at the end. This is a canonical application of the stack abstract data type within data structures, and the same procedure underlies the delimiter-matching check a compiler performs during parsing.
An expression's parentheses, braces, and brackets are balanced when every opener has a closer of the same type to its right, every closer has an opener of the same type to its left, and no delimiter …