Program for nth Catalan Number using Recursion

Program for nth Catalan Number using Dynamic Programming:

Catalan numbers satisfy the following recursive formula:

Step-by-step approach:

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.

Following is the implementation of the above recursive formula.

C++

#include <iostream>
using namespace std;
 
// A recursive function to find nth catalan number
unsigned long int catalan(unsigned int n)
{
    // Base case
    if (n <= 1)
        return 1;
 
    // catalan(n) is sum of
    // catalan(i)*catalan(n-i-1)
    unsigned long int res = 0;
    for (int i = 0; i < n; i++)
        res += catalan(i) * catalan(n - i - 1);
 
    return res;
}
 
// Driver code
int main()
{
    for (int i = 0; i < 10; i++)
        cout << catalan(i) << " ";
    return 0;
}

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) + " ");
        }
    }
}

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 Code
for i in range(10):
    print(catalan(i), end=" ")
# This code is contributed by
# Nikhil Kumar Singh (nickzuck_007)

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;
    }
 
    // Driver Code
    public static void Main()
    {
        for (int i = 0; i < 10; i++)
            Console.Write(catalan(i) + " ");
    }
}
 
// This code is contributed by
// nitin mittal.

Javascript

<script>
 
// Javascript Program for nth 
// Catalan Number
 
// A recursive function to
// find nth catalan number
function catalan(n)
{
     
    // Base case
    if (n <= 1)
        return 1;
 
    // catalan(n) is sum of 
    // catalan(i)*catalan(n-i-1)
    let res = 0;
    for(let i = 0; i < n; i++)
        res += catalan(i) * 
                catalan(n - i - 1);
 
    return res;
}
 
// Driver Code
for (let i = 0; i < 10; i++)
    document.write(catalan(i) + " ");
 
// This code is contributed _saurabh_jaiswal
 
</script>

PHP

<?php
// PHP Program for nth 
// Catalan Number
 
// A recursive function to
// find nth catalan number
function catalan($n)
{
     
    // Base case
    if ($n <= 1)
        return 1;
 
    // catalan(n) is sum of 
    // catalan(i)*catalan(n-i-1)
    $res = 0;
    for($i = 0; $i < $n; $i++)
        $res += catalan($i) * 
                catalan($n - $i - 1);
 
    return $res;
}
 
// Driver Code
for ($i = 0; $i < 10; $i++)
    echo catalan($i), " ";
 
// This code is contributed aj_36
?>

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)

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 C_n, 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.

See this for more applications.

Examples:

Input: n = 6
Output: 132

Input: n = 8
Output: 1430

Tags:

#Amazon #catalan #series #DSA #Dynamic Programming #Mathematical #Amazon #Dynamic Programming #Mathematical #series