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, …**

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

__PRACTICE__**Recursive Solution**

Catalan numbers satisfy the following recursive formula.

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. ` |

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**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!

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