Generate All Strings With n Matched Parentheses

We use directed recursion to generate all strings with n matched sets of parentheses.

Table of Contents
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Code Sample

Authorship: ALL credit for the code in this file goes to the authors of the
book "Elements of Programming Interviews" by Adnan Aziz, Amit Prakash, and
Tsung-Hsien Lee.
I have just adapted the code to pass on Leetcode, added explanatory comments,
reformatted the code, & changed variable names for understanding.
Generate Parentheses - LeetCode:
This code passes all Leetcode test cases as of Jan. 13 2019
Runtime: 2 ms, faster than 95.74% of Java online submissions for Generate Parentheses.
The video to explain this code is here:
public static List<String> generateParenthesis(int numPairs) {
List<String> result = new ArrayList<>();
directedGenerateBalancedParentheses(numPairs, numPairs, "", result); // kick off the recursion
return result;
private static void directedGenerateBalancedParentheses(int numLeftParensNeeded , int numRightParensNeeded,
String parenStringInProgress, List<String> result) {
The recursion has bottomed out.
We have used all left and right parens necessary within constraints up
to this point. Therefore, the answer we add will be a valid paren string.
We can add this answer and then backtrack so the previous call can exhaust
more possibilities and express more answers...and then return to its caller,
etc. etc.
Yeah...this is what backtracking is all about.
if (numLeftParensNeeded == 0 && numRightParensNeeded == 0) {
At each frame of the recursion we have 2 things we can do:
1.) Insert a left parenthesis
2.) Insert a right parenthesis
These represent all of the possibilities of paths we can take from this
respective call. The path that we can take all depends on the state coming
into this call.
Can we insert a left parenthesis? Only if we have lefts remaining to insert
at this point in the recursion
if (numLeftParensNeeded > 0) {
numLeftParensNeeded - 1 -> We are using a left paren
numRightParensNeeded -> We did not use a right paren
parenStringInProgress + "(" -> We append a left paren to the string in progress
result -> Just pass the result list along for the next call to use
directedGenerateBalancedParentheses(numLeftParensNeeded - 1, numRightParensNeeded,
parenStringInProgress + "(", result);
Can we insert a right parenthesis? Only if the number of left parens needed
is less than then number of right parens needed.
This means that there are open left parenthesis to close OTHERWISE WE CANNOT
USE A RIGHT TO CLOSE ANYTHING. We would lose balance.
if (numLeftParensNeeded < numRightParensNeeded) {
numLeftParensNeeded -> We did not use a left paren
numRightParensNeeded - 1 -> We used a right paren
parenStringInProgress + ")" -> We append a right paren to the string in progress
result -> Just pass the result list along for the next call to use
directedGenerateBalancedParentheses(numLeftParensNeeded, numRightParensNeeded - 1,
parenStringInProgress + ")", result);