In this article, we will learn about calculating the nth Catalan number.
Catalan numbers are a sequence of natural numbers that are defined by the recursive formula −
$$c_{0} = 1\;and\; c_{n+1} = \displaystyle\sum\limits_{i=0}^nc_{i} c_{n-i}\; for n\geq 0 ;$$
The first few Catalan numbers for n = 0, 1, 2, 3, … are 1, 1, 2, 5, 14, 42, 132, 429,...................
Catalan numbers can be obtained both by recursion and dynamic programming.
So let’s see their implementation.
Approach 1: Recursion Method
Example
Approach 1: Recursion Method
# A recursive solution def catalan(n): #negative value if n <=1 : return 1 # Catalan(n) = catalan(i)*catalan(n-i-1) res = 0 for i in range(n): res += catalan(i) * catalan(n-i-1) return res # main for i in range(6): print (catalan(i))
Output
1 1 2 5 14 42
The scope of all the variables and recursive calls are shown below.
Approach 2: Dynamic Programming Method
Example
# using dynamic programming def catalan(n): if (n == 0 or n == 1): return 1 # divide table catalan = [0 for i in range(n + 1)] # Initialization catalan[0] = 1 catalan[1] = 1 # recursion for i in range(2, n + 1): catalan[i] = 0 for j in range(i): catalan[i] = catalan[i] + catalan[j] * catalan[i-j-1] return catalan[n] # main for i in range (6): print (catalan(i),end=" ")
Output
1 1 2 5 14 42
The scope of all the variables and recursive calls are shown below.
Conclusion
In this article, we learned about the method of generating the nth Catalan number