[Leetcode] Generate Parenthesis

题目：

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

For example, given n = 3, a solution set is:

"((()))", "(()())", "(())()", "()(())", "()()()"

思路：递归列举所有可能，同时分别记录( 和 ) 的数量。

class Solution {
public:
    string trace;
    
    void generate_parenthesis_helper(int sum, int left_sum, int right_sum, vector<string>& result) {
        if (left_sum == sum && right_sum == sum) {
            result.push_back(trace);
            return;
        }
        if (left_sum < sum) {   //as long as left_sum < sum, always safe to generate '('
            trace.push_back('(');
            generate_parenthesis_helper(sum, left_sum + 1, right_sum, result);
            trace.pop_back();
        }
        if (right_sum < left_sum) {   //as long as right_sum < left_sum, we can generate ')'
            trace.push_back(')');
            generate_parenthesis_helper(sum, left_sum, right_sum + 1, result);
            trace.pop_back();
        }
    }
    
    vector<string> generateParenthesis(int n) {
        vector<string> result;
        generate_parenthesis_helper(n, 0, 0, result);
        return result;
    }
};