Java Program for nth Catalan Number using Recursive Solution
Catalan numbers satisfy the following recursive formula:
Follow the steps below to implement the above recursive formula
- Base condition for the recursive approach, when n <= 1, return 1
- Iterate from i = 0 to i < n
- Make a recursive call catalan(i) and catalan(n – i – 1) and keep adding the product of both into res.
- Return the res.
Below is the implementation of the above approach:
Java
import java.io.*;class CatalnNumber { // A recursive function to find nth catalan number 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; } // Driver Code public static void main(String[] args) { CatalnNumber cn = new CatalnNumber(); for (int i = 0; i < 10; i++) { System.out.print(cn.catalan(i) + " "); } }} |
Output
1 1 2 5 14 42 132 429 1430 4862
Time Complexity: The above implementation is equivalent to nth Catalan number.
The value of nth Catalan number is exponential which makes the time complexity exponential.
Auxiliary Space: O(n)
Java Program for nth Catalan Number
Catalan numbers are defined as a mathematical sequence that consists of positive integers, which can be used to find the number of possibilities of various combinations.
The nth term in the sequence denoted Cn, is found in the following formula:
The first few Catalan numbers for n = 0, 1, 2, 3, … are : 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …
Catalan numbers occur in many interesting counting problems like the following.
- Count the number of expressions containing n pairs of parentheses that are correctly matched. For n = 3, possible expressions are ((())), ()(()), ()()(), (())(), (()()).
- Count the number of possible Binary Search Trees with n keys (See this)
- Count the number of full binary trees (A rooted binary tree is full if every vertex has either two children or no children) with n+1 leaves.
- Given a number n, return the number of ways you can draw n chords in a circle with 2 x n points such that no 2 chords intersect.
Examples:
Input: n = 6
Output: 132Input: n = 8
Output: 1430