Q:

Java program to find the total number of possible Binary Search Trees with N keys

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robaa
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2022-11-13
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In this program, we need to find out the total number of binary search trees can be constructed with n values. Below diagram shows a possible binary search tree with the key value as 3. So, we can construct a total of five binary search trees. When we choose node 1 as the root node, we get two trees. Similarly, one tree with 2 as root nodes and two trees when we select 3 as the root node.

This approach involves selecting a node recursively as the root node and create possible binary search tree.

An easy way to calculate the total number of possible binary search trees are through Catalan number:

 

Cn = (2n)! / n! *(n+1)!  

Algorithm

  • Define Node class which has three attributes namely: data left and right. Here, left represents the left child of the node and right represents the right child of the node.
  • When a node is created, data will be passed to the data attribute of the node and both left and right will be set to null.
  • Define another class which has an attribute root.
    • Root represents the root node of the tree and initializes it to null.

a. numOfBST() will find out total possible binary search tree for given key:

 

  • It will calculate the Catalan number for given key by making a call to factorial().
  • Catalan number can be calculated using the formula:
    Cn = (2n)! / n! *(n+1)!
  • Factorial() will calculate factorial of a given number.
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robaa
2022-11-13
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Program:

public class BinarySearchTree {  
  
    //Represent the node of binary tree  
    public static class Node{  
        int data;  
        Node left;  
        Node right;  
  
        public Node(int data){  
            //Assign data to the new node, set left and right children to null  
            this.data = data;  
            this.left = null;  
            this.right = null;  
            }  
        }  
  
    //Represent the root of binary tree  
    public Node root;  
  
    public BinarySearchTree(){  
        root = null;  
    }  
  
    //factorial() will calculate the factorial of given number  
    public int factorial(int num) {  
        int fact = 1;  
        if(num == 0)  
            return 1;  
        else {  
            while(num > 1) {  
                fact = fact * num;  
                num--;  
            }  
            return fact;  
        }  
    }  
  
    //numOfBST() will calculate the total number of possible BST by calculating Catalan Number for given key  
    public int numOfBST(int key) {  
        int catalanNumber = factorial(2 * key)/(factorial(key + 1) * factorial(key));  
        return catalanNumber;  
    }  
  
    public static void main(String[] args) {  
  
        BinarySearchTree bt = new BinarySearchTree();  
  
        //Display total number of possible binary search tree with key 5  
        System.out.println("Total number of possible Binary Search Trees with given key: " + bt.numOfBST(5));  
      }  
}

Output:

Total number of possible Binary Search Trees with given key: 42

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