Generate Parentheses

Given n pairs of parentheses, write a function to generate all combinations of well-formed parentheses.

Example: n = 3
Output: ["((()))","(()())","(())()","()(())","()()()"]

Instead of generating all $2^{2n}$ strings and validating them, we can build only valid strings.

Rules:

1. Open: We can add an open parenthesis ( if open_count < n.
2. Close: We can add a closing parenthesis ) if closed_count < open_count.

This ensures we never close more than we open (validity) and never exceed n (completeness).

Visualize the decision choices between '(' and ')'. Red paths would violate validity constraints and are pruned.