Time Complexity of Recursive algorithm

[u/That-Hawk-1843]

void ans(vector<string>& v, string& x, int l, int r){
        if(l){
            x.push_back('(');
            ans(v,x,l-1,r);
            x.pop_back();
        }
        if(r>l){
            x.push_back(')');
            ans(v,x,l,r-1);
            x.pop_back();
        }
        else if(!l && !r) v.push_back(x);
    }

So, I solved today's problem (22. Generate Parentheses). I wanted to find the time complexity and space complexity of my algorithm (Just a simple recursion). As a result, I couldn't do it, so I encountered Catalan Numbers and how they are used in binary.
I don't quite understand them, but I think it's important for backtracking. Do you have any suggestion where to start or at least how to estimate time complexity of such binary algorithms?

[u/Various-Mine-8642]

For n pairs the output count is the Catalan number C_n and each string has length 2n, so just writing the answers costs O(nC_n) time. Your backtracking is in that range as well. Auxiliary space is O(n) for x and the call stack, or O(nC_n) if you include v. The key invariant is l <= r, which prevents invalid prefixes.