Understanding the usecase of multiple recursive calls

Question

I was trying to solve a problem to get all the possible valid combinations of parenthesis given an integer. Eg. input: n = 2, output: (()), ()()

Clearly as n increases, the current output builds on the output of the previous n. So it was easy to come up with a recursive by taking the previous result and adding to it to get the current result:

HashSet<String> getCombinations(int n)
{
    HashSet<String> result = new HashSet<String>();
    if (n <= 0) return null;
    if (n == 1)
    {
        result.add("()");
        return result;
    }
    for (String str: getCombinations(n - 1))
    {
        for (int i = 0; i < str.length(); i++)
        {
            result.add(new StringBuffer(str).insert(i, "()").toString());
        }
    }
    return result;
}

Though obviously the downside of the above code is the repetition of the same result values which are produced but not stored. So I looked online for a better solution (as I could not think of it), and found this:

ArrayList<String> result = new ArrayList<>();
void getCombinations(int index, int nLeft, int nRight, String str)
{
    if (nLeft < 0 || nRight < 0 || nLeft > nRight) return;
    if (nLeft == 0 && nRight == 0)
    {
        result.add(str);
        return;
    }
    getCombinations(index + 1, nLeft - 1, nRight, new StringBuffer(str).insert(index, "(").toString());
    getCombinations(index + 1, nLeft, nRight - 1, new StringBuffer(str).insert(index, ")").toString());
}

I understand how this solution works and why its better than the first. But even now I cannot imagine looking at the first solution and then coming up with second solution. How can I intuitively understand so as to when to use multiple recursive calls? In other words, after achieving solution 1, how can I come to think that I would probably be better off with multiple recursive calls?

My question is not specific to the above problem, but the type of problems in general.

Answer 1

You could look at this problem as permutations of 1 and -1, which while being summed together, the running sum (temp value while adding up numbers left-to-right) must not become less than 0, or in case of parenthesis, must not use more right-parenthesis than left ones. So:

If n=1 , then you can only do 1+(-1) . If n=2 , then can do 1+(-1)+1+(-1) or 1+1+(-1)+(-1) .

So, as you create these permutations, you see you can only use only one of the two options - either 1 or -1 - in every recursive call, and by keeping track of what has been used with nLeft and nRight you can have program know when to stop.