C# Program for nth Catalan Number

Catalan numbers are a sequence of natural numbers that occurs in many interesting counting problems like following.

1) Count the number of expressions containing n pairs of parentheses which are correctly matched. For n = 3, possible expressions are ((())), ()(()), ()()(), (())(), (()()).

2) Count the number of possible Binary Search Trees with n keys (See this)
See this for more applications.

The first few Catalan numbers for n = 0, 1, 2, 3, … are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Recursive Solution
Catalan numbers satisfy the following recursive formula.
$C_0=1 \ and \ C_n_+_1=\sum_{i=0}^{n}C_iC_n_-_i \ for \ n\geq 0;$
Following is the implementation of above recursive formula.

C#

// A recursive C# program to find 
// nth catalan number 
using System; 
  
class GFG { 
  
    // A recursive function to find 
    // nth catalan number 
    static int catalan(int n) 
    { 
        int res = 0; 
  
        // Base case 
        if (n <= 1) { 
            return 1; 
        } 
        for (int i = 0; i < n; i++) { 
            res += catalan(i) 
                   * catalan(n - i - 1); 
        } 
        return res; 
    } 
  
    public static void Main() 
    { 
        for (int i = 0; i < 10; i++) 
            Console.Write(catalan(i) 
                          + " "); 
    } 
} 
  
// This code is contributed by 
// nitin mittal. 

Output:

1 1 2 5 14 42 132 429 1430 4862

Dynamic Programming Solution
We can observe that the above recursive implementation does a lot of repeated work (we can the same by drawing recursion tree). Since there are overlapping subproblems, we can use dynamic programming for this. Following is a Dynamic programming based implementation in C++.

Please refer complete article on Program for nth Catalan Number for more details!

My Personal Notes

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

C#

Recommended Posts: