Python Program for nth Catalan Number

Catalan numbers are a sequence of natural numbers that occurs in many interesting counting problems like the 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, …

Recursive Solution

Catalan numbers satisfy the following recursive formula. 

Following is the implementation of above recursive formula.  

Python3

# A recursive function to find nth catalan number
def catalan(n):
    # Base Case
    if n <= 1:
        return 1

    # Catalan(n) is the sum of catalan(i)*catalan(n-i-1)
    res = 0
    for i in range(n):
        res += catalan(i) * catalan(n-i-1)

    return res


# Driver Program to test above function
for i in range(10):
    print(catalan(i), end=' ')

Output:

1 1 2 5 14 42 132 429 1430 4862

Time Complexity: O(2ⁿ)

Auxiliary Space:  O(n)

Dynamic Programming Solution 

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

Python3

# A dynamic programming based function to find nth
# Catalan number

def catalan(n):
    if (n == 0 or n == 1):
        return 1

    # Table to store results of subproblems
    catalan = [0 for i in range(n + 1)]

    # Initialize first two values in table
    catalan[0] = 1
    catalan[1] = 1

    # Fill entries in catalan[] using recursive formula
    for i in range(2, n + 1):
        catalan[i] = 0
        for j in range(i):
            catalan[i] += catalan[j] * catalan[i-j-1]

    # Return last entry
    return catalan[n]


# Driver code
for i in range(10):
    print(catalan(i))

Output:

1
1
2
5
14
42
132
429
1430
4862

Time Complexity: O(n^2)
Auxiliary Space: O(n)

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