C/C++ Program for nth Catalan Number?

Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. The nth Catalan number can be calculated using the recursive formula or dynamic programming approaches.

Syntax

C(n) = (2n)! / ((n+1)! * n!)
C(n) = C(0)*C(n-1) + C(1)*C(n-2) + ... + C(n-1)*C(0)

Mathematical Properties

Catalan numbers have several interpretations −

• C_n is the number of Dyck words of length 2n (strings with n X's and n Y's where no prefix has more Y's than X's)
• C_n counts valid parentheses combinations with n pairs
• C_n represents ways to parenthesize n+1 factors completely
• C_n is the number of full binary trees with n+1 leaves

Method 1: Recursive Approach

The simplest approach uses the recursive formula where each Catalan number is the sum of products of previous Catalan numbers −

#include <stdio.h>

long int catalan(int n) {
    if (n <= 1) {
        return 1;
    }
    
    long int result = 0;
    for (int i = 0; i < n; i++) {
        result += catalan(i) * catalan(n - i - 1);
    }
    return result;
}

int main() {
    printf("First 6 Catalan numbers: ");
    for (int i = 0; i < 6; i++) {
        printf("%ld ", catalan(i));
    }
    printf("\n");
    return 0;
}

First 6 Catalan numbers: 1 1 2 5 14 42 

Method 2: Dynamic Programming

For better efficiency, we can use dynamic programming to avoid recalculating the same values −

#include <stdio.h>

long int catalanDP(int n) {
    if (n <= 1) return 1;
    
    long int catalan[n + 1];
    catalan[0] = catalan[1] = 1;
    
    for (int i = 2; i <= n; i++) {
        catalan[i] = 0;
        for (int j = 0; j < i; j++) {
            catalan[i] += catalan[j] * catalan[i - j - 1];
        }
    }
    return catalan[n];
}

int main() {
    int n = 10;
    printf("Catalan numbers from 0 to %d:\n", n);
    for (int i = 0; i <= n; i++) {
        printf("C(%d) = %ld\n", i, catalanDP(i));
    }
    return 0;
}

Catalan numbers from 0 to 10:
C(0) = 1
C(1) = 1
C(2) = 2
C(3) = 5
C(4) = 14
C(5) = 42
C(6) = 132
C(7) = 429
C(8) = 1430
C(9) = 4862
C(10) = 16796

Comparison

Conclusion

Catalan numbers can be computed using recursive formulas or dynamic programming. The DP approach is more efficient for larger values, avoiding redundant calculations with O(n²) complexity instead of exponential time.