Generate each parenthesis string from left to right. At any prefix, the number of closing parentheses cannot exceed the number of opening parentheses; otherwise no suffix could make the string balanced. Also, a complete balanced string of length N has exactly N/2 opening parentheses.
The recursion therefore adds ( while fewer than N/2 openings have been used, and adds ) only when close < open. A prefix with equal counts is allowed: subsequent openings can still make it valid. When the length reaches N, these rules guarantee the string is balanced, so it can be emitted. Trying ( before ) visits the valid strings in lexicographic order. For odd N, no leaf can satisfy the equal-count condition, so no result is printed.
The prefix-balance invariant is 0 <= close <= open <= N/2. Each result is generated once, and the work is proportional to the total output size; the recursion and current string use O(N) working space, in addition to the buffered output.
Java
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import java.io.BufferedInputStream;
import java.io.IOException;
public class Main {
private static int n;
private static final StringBuilder current = new StringBuilder();
private static final StringBuilder output = new StringBuilder();
public static void main(String[] args) throws IOException {
FastScanner input = new FastScanner();
n = input.nextInt();
generate(0, 0);
System.out.print(output);
}
private static void generate(int open, int close) {
if (current.length() == n) {
if (open == close) {
output.append(current).append('\n');
}
return;
}
if (open < n / 2) {
current.append('(');
generate(open + 1, close);
current.setLength(current.length() - 1);
}
if (close < open) {
current.append(')');
generate(open, close + 1);
current.setLength(current.length() - 1);
}
}
private static final class FastScanner {
private final BufferedInputStream input = new BufferedInputStream(System.in);
int nextInt() throws IOException {
int value = 0;
int c;
do {
c = input.read();
} while (c <= ' ' && c != -1);
while (c > ' ') {
value = value * 10 + c - '0';
c = input.read();
}
return value;
}
}
}