Catalan Number in Java

In this section, we will learn what is a Catalan number and also create Java programs to check if the given number is a Catalan number or not. The Catalan number program is frequently asked in Java coding interviews and academics.

There are many interesting problems that can be solved using the Catalan number. Mathematically, the Catalan numbers are defined as,

Steps to Find the Catalan Numbers

Step 1: Assign a non-negative integer to the variable n.

Step 2: Find the value of ²ⁿC_n, where n is determined in step 1.

Step 3: Divide the value found in step 2 by n+1. The resultant that we get after the division is a Catalan number.

Example Catalan Number

The following diagram shows Catalan numbers for numbers from 0 to 3.

Catalan Number Java Program

The following program is based on the steps defined above.

FileName: CatalanNumberExample.java

// A Java program to calculate the Catalan numbers
public class CatalanNumberExample 
{

// a method that finds the factorial of a number 
public long findFactorial(int no)
{
long res = 1;

// calculating the factorial
for(int i = 1; i <= no; i++)
{
res = res * i;
}

return res;
}

// A method that finds the Catalan number corresponding to 
// the number no
public long findCatalan(int no)
{

// for no = 0 & 1, the catalan number is 1
if(no <= 1) { return 1; }

// calculating 2n!
long fact = findFactorial(2 * no);

// calculating 2nCn
fact = fact / findFactorial(no);
fact = fact / findFactorial(no);

// caculating ((1 / (n + 1))* 2nCn)
fact = fact / (no + 1);

// returning the calculated Catalan number
return fact;
}

// main method
public static void main(String[] argvs)
{
// Creating an object of the class CatalanNumberExample
CatalanNumberExample obj = new CatalanNumberExample();

// Calculating catalan numbers from 0 to 10
for (int i = 0; i <= 10; i++) 
{
long catalanNo = obj.findCatalan(i);
System.out.println("For n = " + i + ", Catalan number is " + catalanNo);
}
}
}

Output:

For n = 0, Catalan number is 1
For n = 1, Catalan number is 1
For n = 2, Catalan number is 2
For n = 3, Catalan number is 5
For n = 4, Catalan number is 14
For n = 5, Catalan number is 42
For n = 6, Catalan number is 132
For n = 7, Catalan number is 429
For n = 8, Catalan number is 1430
For n = 9, Catalan number is 4862
For n = 10, Catalan number is 16796

Catalan Numbers Using Recursive Approach

We can also find the Catalan number by using the recursive approach. It can be defined as follows.

Similarly, we can find other values too. Note that while finding the Catalan number for the next term, use the values of the previous terms. For example, the value of C(4) is dependent on C(0), C(1), C(2), and C(3).

Let's implement the recursive approach in a Java program.

FileName: CatalanNumberExample1.java

// A Java program to calculate the Catalan numbers from 0 to 10
public class CatalanNumberExample1 
{
 
// A method that returns the Catalan number for the number no  
int findCatalanNo(int no)
{
    // base case
    if(no <= 1)
    {
        return 1;
    }
    
    // for storing the catalan number
    int res = 0;
    for (int j = 0; j < no; j++)
    {
        // recursively find the catalan number
        res = res + findCatalanNo(j) * findCatalanNo(no - j - 1);
    }
    
    return res;
}

// main method
public static void main(String argvs[])
{
// creating an object of the class CatalanNumberExample1
CatalanNumberExample1 obj = new CatalanNumberExample1();

for(int i = 0; i <= 10; i++)
{
// find Catalan number for each value of i
int no = obj.findCatalanNo(i);

System.out.println("For number = " + i + ", the Catalan number is " + no);
}
}
}

Output:

For number = 0, the Catalan number is 1
For number = 1, the Catalan number is 1
For number = 2, the Catalan number is 2
For number = 3, the Catalan number is 5
For number = 4, the Catalan number is 14
For number = 5, the Catalan number is 42
For number = 6, the Catalan number is 132
For number = 7, the Catalan number is 429
For number = 8, the Catalan number is 1430
For number = 9, the Catalan number is 4862
For number = 10, the Catalan number is 16796

Using Dynamic Programming

The recursive program written above takes a lot of time to find the Catalan numbers. It is because we are using recursion inside the Java for-loop. Therefore, we need to find another approach that takes less time as compared to the above one.

By using dynamic programming, we can reduce time. In this approach, we will be shifting from recursive to iterative approach, as the iterative approach is much faster. Observe the following program for the same.

FileName: CatalanNumberExample2.java

// A Java program to calculate the Catalan numbers from 1 to 10
public class CatalanNumberExample2
{
// an integer array for storing the Catalan numbers
int arr[];
// constructor of the class
CatalanNumberExample1(int n)
{
// we have to store Catalan numbers from 0 to n; hence, n + 1
arr = new int[n + 1];
}
// A method that returns the Catalan number for the number no  
int findCatalanNo(int no)
{
     // for no = 0 & 1, Catalan number is 1
    arr[0] = 1;
    arr[1] = 1;
    // base case
    if(no <= 1)
    {
        return arr[no];
    }    
    // for storing the Catalan number
    int res = 0;
    for (int i = 1; i <= no; i++)
    {
        res = res + arr[i - 1] * arr[no - i];
    }
    
    // updating the array    
    arr[no] = res;
    
    // returning the calculated Catalan number
    return res;
}

// main method
public static void main(String argvs[])
{
// creating an object of the class CatalanNumberExample1
// to find the Catalan numbers from 0 to 10
CatalanNumberExample2 obj = new CatalanNumberExample2(10);
for(int i = 0; i <= 10; i++)
{
// find Catalan number for each value of i
int no = obj.findCatalanNo(i);
System.out.println("For number = " + i + ", the Catalan number is " + no);
}
}
}

Output:

For number = 0, the Catalan number is 1
For number = 1, the Catalan number is 1
For number = 2, the Catalan number is 2
For number = 3, the Catalan number is 5
For number = 4, the Catalan number is 14
For number = 5, the Catalan number is 42
For number = 6, the Catalan number is 132
For number = 7, the Catalan number is 429
For number = 8, the Catalan number is 1430
For number = 9, the Catalan number is 4862
For number = 10, the Catalan number is 16796

Explanation: The problem with the recursive approach was that it was calculating the same subproblem again and again. It leads to more time consumption. Therefore, we have used an array, arr, to store the solution of the subproblem. Hence, whenever there is a need of the solution of a subproblem, we can used it from the array arr. Thus, we have saved the computational time of repeated subproblems.

Application of Catalan Numbers

As discussed above, one can solve many interesting problems using Catalan numbers a few of them are mentioned below.

1) For n > 0, the number of unique Binary Search Trees is equal to the Catalan number C(n), where n is the number of nodes present in the Binary Search Trees.

2) For n > 0, the total number of n pair of parentheses that are correctly matched is equal to the Catalan number C(n).

3) The total number of chords drawn on a circle using 2 × n points such that no two chords intersect or touch each other is given by the Catalan number C(n).