Program for nth Catalan Number

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)
3) 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.
4) 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. 
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. 

filter_none

edit
close

play_arrow

link
brightness_4
code

#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;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

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) + " ");
        }
    }
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# 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),
# This code is contributed by 
# Nikhil Kumar Singh (nickzuck_007)
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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.
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?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
?>
chevron_right

Output

1 1 2 5 14 42 132 429 1430 4862 

Time complexity of above implementation is equivalent to nth catalan number. 

The value of nth catalan number is exponential that makes the time complexity exponential.

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 .
 

filter_none

edit
close

play_arrow

link
brightness_4
code

#include <iostream>
using namespace std;
  
// A dynamic programming based function to find nth
// Catalan number
unsigned long int catalanDP(unsigned int n)
{
    // Table to store results of subproblems
    unsigned long int catalan[n + 1];
  
    // Initialize first two values in table
    catalan[0] = catalan[1] = 1;
  
    // Fill entries in catalan[] using recursive formula
    for (int i = 2; i <= n; i++) 
    {
        catalan[i] = 0;
        for (int j = 0; j < i; j++)
            catalan[i] += catalan[j] 
                       * catalan[i - j - 1];
    }
  
    // Return last entry
    return catalan[n];
}
  
// Driver code
int main()
{
    for (int i = 0; i < 10; i++)
        cout << catalanDP(i) << " ";
    return 0;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

class GFG {
  
    // A dynamic programming based function to find nth
    // Catalan number
    static int catalanDP(int n)
    {
        // Table to store results of subproblems
        int catalan[] = new int[n + 2];
  
        // Initialize first two values in table
        catalan[0] = 1;
        catalan[1] = 1;
  
        // Fill entries in catalan[] 
        // using recursive formula
        for (int i = 2; i <= n; i++) 
        {
            catalan[i] = 0;
            for (int j = 0; j < i; j++) 
            {
                catalan[i]
                    += catalan[j] 
                    * catalan[i - j - 1];
            }
        }
  
        // Return last entry
        return catalan[n];
    }
  
    // Driver code
    public static void main(String[] args)
    {
        for (int i = 0; i < 10; i++) {
            System.out.print(catalanDP(i) + " ");
        }
    }
}
// This code contributed by Rajput-Ji
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# 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] *= catalan[i-j-1]
  
    # Return last entry
    return catalan[n]
  
  
# Driver code
for i in range(10):
    print(catalan(i), end=" ")
# This code is contributed by Aditi Sharma
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

using System;
  
class GFG {
  
    // A dynamic programming based
    // function to find nth
    // Catalan number
    static uint catalanDP(uint n)
    {
        // Table to store results of subproblems
        uint[] catalan = new uint[n + 2];
  
        // Initialize first two values in table
        catalan[0] = catalan[1] = 1;
  
        // Fill entries in catalan[]
        // using recursive formula
        for (uint i = 2; i <= n; i++) {
            catalan[i] = 0;
            for (uint j = 0; j < i; j++)
                catalan[i]
                    += catalan[j] * catalan[i - j - 1];
        }
  
        // Return last entry
        return catalan[n];
    }
  
    // Driver code
    static void Main()
    {
        for (uint i = 0; i < 10; i++)
            Console.Write(catalanDP(i) + " ");
    }
}
  
// This code is contributed by Chandan_jnu
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program for nth Catalan Number
  
// A dynamic programming based function 
// to find nth Catalan number
function catalanDP( $n)
{
      
    // Table to store results 
    // of subproblems
    $catalan= array();
  
    // Initialize first two 
    // values in table
    $catalan[0] = $catalan[1] = 1;
  
    // Fill entries in catalan[] 
    // using recursive formula
    for ($i = 2; $i <= $n; $i++)
    {
        $catalan[$i] = 0;
        for ( $j = 0; $j < $i; $j++)
            $catalan[$i] += $catalan[$j] * 
                   $catalan[$i - $j - 1];
    }
  
    // Return last entry
    return $catalan[$n];
}
  
    // Driver Code
    for ($i = 0; $i < 10; $i++)
        echo catalanDP($i) , " ";
  
// This code is contributed anuj_67.
?>
chevron_right

Output
1 1 2 5 14 42 132 429 1430 4862 

Time Complexity: Time complexity of above implementation is O(n2)

Using Binomial Coefficient 
We can also use the below formula to find nth Catalan number in O(n) time. 
 



We have discussed a O(n) approach to find binomial coefficient nCr

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program for nth Catalan Number
#include <iostream>
using namespace std;
  
// Returns value of Binomial Coefficient C(n, k)
unsigned long int binomialCoeff(unsigned int n,
                                unsigned int k)
{
    unsigned long int res = 1;
  
    // Since C(n, k) = C(n, n-k)
    if (k > n - k)
        k = n - k;
  
    // Calculate value of [n*(n-1)*---*(n-k+1)] /
    // [k*(k-1)*---*1]
    for (int i = 0; i < k; ++i) {
        res *= (n - i);
        res /= (i + 1);
    }
  
    return res;
}
  
// A Binomial coefficient based function to find nth catalan
// number in O(n) time
unsigned long int catalan(unsigned int n)
{
    // Calculate value of 2nCn
    unsigned long int c = binomialCoeff(2 * n, n);
  
    // return 2nCn/(n+1)
    return c / (n + 1);
}
  
// Driver code
int main()
{
    for (int i = 0; i < 10; i++)
        cout << catalan(i) << " ";
    return 0;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program for nth Catalan Number
  
class GFG {
  
    // Returns value of Binomial Coefficient C(n, k)
    static long binomialCoeff(int n, int k)
    {
        long res = 1;
  
        // Since C(n, k) = C(n, n-k)
        if (k > n - k) {
            k = n - k;
        }
  
        // Calculate value of [n*(n-1)*---*(n-k+1)] /
        // [k*(k-1)*---*1]
        for (int i = 0; i < k; ++i) {
            res *= (n - i);
            res /= (i + 1);
        }
  
        return res;
    }
  
    // A Binomial coefficient based function
    //  to find nth catalan number in O(n) time
    static long catalan(int n)
    {
        // Calculate value of 2nCn
        long c = binomialCoeff(2 * n, n);
  
        // return 2nCn/(n+1)
        return c / (n + 1);
    }
  
    // Driver code
    public static void main(String[] args)
    {
        for (int i = 0; i < 10; i++) {
            System.out.print(catalan(i) + " ");
        }
    }
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python program for nth Catalan Number
# Returns value of Binomial Coefficient C(n, k)
  
  
def binomialCoefficient(n, k):
  
    # since C(n, k) = C(n, n - k)
    if (k > n - k):
        k = n - k
  
    # initialize result
    res = 1
  
    # Calculate value of [n * (n-1) *---* (n-k + 1)]
    # / [k * (k-1) *----* 1]
    for i in range(k):
        res = res * (n - i)
        res = res / (i + 1)
    return res
  
# A Binomial coefficient based function to
# find nth catalan number in O(n) time
  
  
def catalan(n):
    c = binomialCoefficient(2*n, n)
    return c/(n + 1)
  
# Driver Code
for i in range(10):
    print(catalan(i), end=" ")
  
# This code is contributed by Aditi Sharma
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program for nth Catalan Number
using System;
class GFG {
  
    // Returns value of Binomial Coefficient C(n, k)
    static long binomialCoeff(int n, int k)
    {
        long res = 1;
  
        // Since C(n, k) = C(n, n-k)
        if (k > n - k) {
            k = n - k;
        }
  
        // Calculate value of [n*(n-1)*---*(n-k+1)] /
        // [k*(k-1)*---*1]
        for (int i = 0; i < k; ++i) {
            res *= (n - i);
            res /= (i + 1);
        }
  
        return res;
    }
  
    // A Binomial coefficient based function to find nth
    // catalan number in O(n) time
    static long catalan(int n)
    {
        // Calculate value of 2nCn
        long c = binomialCoeff(2 * n, n);
  
        // return 2nCn/(n+1)
        return c / (n + 1);
    }
  
    // Driver code
    public static void Main()
    {
        for (int i = 0; i < 10; i++) {
            Console.Write(catalan(i) + " ");
        }
    }
}
  
// This code is contributed
// by Akanksha Rai
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program for nth Catalan Number 
  
// Returns value of Binomial 
// Coefficient C(n, k) 
function binomialCoeff($n, $k
    $res = 1; 
  
    // Since C(n, k) = C(n, n-k) 
    if ($k > $n - $k
        $k = $n - $k
  
    // Calculate value of [n*(n-1)*---*(n-k+1)] / 
    //                    [k*(k-1)*---*1] 
    for ($i = 0; $i < $k; ++$i
    
        $res *= ($n - $i); 
        $res = floor($res / ($i + 1)); 
    
  
    return $res
  
// A Binomial coefficient based function 
// to find nth catalan number in O(n) time 
function catalan($n
    // Calculate value of 2nCn 
    $c = binomialCoeff(2 * ($n), $n); 
  
    // return 2nCn/(n+1) 
    return floor($c / ($n + 1)); 
  
// Driver code 
for ($i = 0; $i < 10; $i++) 
echo catalan($i), " "
  
// This code is contributed by Ryuga
?>
chevron_right

Output
1 1 2 5 14 42 132 429 1430 4862 

Time Complexity: Time complexity of above implementation is O(n).
We can also use below formula to find nth catalan number in O(n) time. 

Use multi-precision library:  In this method, we have used boost multi-precision library, and the motive behind its use is just only to have precision meanwhile finding the large CATALAN’s number and a generalized technique using for loop to calculate Catalan numbers .  

For example: N = 5

Initially set cat_=1 then, print cat_  ,

then, iterate from i = 1 to i < 5

for i = 1; cat_ = cat_ * (4*1-2)=1*2=2
cat_ = cat_ / (i+1)=2/2 = 1      

For i = 2; cat_ = cat_ * (4*2-2)=1*6=6
cat_ = cat_ / (i+1)=6/3=2  



For i = 3 :-      cat_ = cat_ * (4*3-2)=2*10=20
cat_ = cat_ / (i+1)=20/4=5   

For i = 4 :-      cat_ = cat_ * (4*4-2)=5*14=70
 cat_ = cat_ / (i+1)=70/5=14     

Pseudocode: 

a) initially set cat_=1 and print it
b) run a for loop i=1 to i<=n
            cat_ *= (4*i-2)
            cat_ /= (i+1)
            print cat_
c) end loop and exit        
filter_none

edit
close

play_arrow

link
brightness_4
code

#include <bits/stdc++.h>
#include <boost/multiprecision/cpp_int.hpp>
using boost::multiprecision::cpp_int;
using namespace std;
  
// Function to print the number
void catalan(int n)
{
    cpp_int cat_ = 1;
  
    // For the first number
    cout << cat_ << " "; // C(0)
  
    // Iterate till N
    for (cpp_int i = 1; i < n; i++) 
    {
        // Calculate the number
        // and print it
        cat_ *= (4 * i - 2);
        cat_ /= (i + 1);
        cout << cat_ << " ";
    }
}
  
// Driver code
int main()
{
    int n = 5;
  
    // Function call
    catalan(n);
    return 0;
}
chevron_right

Output
1 1 2 5 14 

Time Complexity: O(n)
Auxiliary Space: O(1)

Another solution using BigInteger in java:

filter_none

edit
close

play_arrow

link
brightness_4
code

import java.io.*;
import java.util.*;
import java.math.*;
  
class GFG 
{
    public static BigInteger findCatalan(int n)
    {
        // using BigInteger to calculate large factorials
        BigInteger b = new BigInteger("1");
  
        // calculating n!
        for (int i = 1; i <= n; i++) {
            b = b.multiply(BigInteger.valueOf(i));
        }
  
        // calculating n! * n!
        b = b.multiply(b);
  
        BigInteger d = new BigInteger("1");
  
        // calculating (2n)!
        for (int i = 1; i <= 2 * n; i++) {
            d = d.multiply(BigInteger.valueOf(i));
        }
  
        // calculating (2n)! / (n! * n!)
        BigInteger ans = d.divide(b);
  
        // calculating (2n)! / ((n! * n!) * (n+1))
        ans = ans.divide(BigInteger.valueOf(n + 1));
        return ans;
    }
    
    // Driver Code
    public static void main(String[] args)
    {
        int n = 5;
        System.out.println(findCatalan(n));
    }
}
// Contributed by Rohit Oberoi
chevron_right

Output
42



References: 
http://en.wikipedia.org/wiki/Catalan_number 
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
 

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.





Article Tags :