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Author: MadyDixit

data_structures/binary_tree/N_Possible_BST.py

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+'''
+Hey, We Are Going to find Some Exciting Number Called Catalan Number which is use to find the Possible binary search
+tree and Binary Search Tree from the Number of Nodes.
+WE USE THE FORMULA:
+t(n) = SUMMATION(i = 1 to n)t(i-1)t(n-i)
+
+You can get Further Detail from Wikipedia:
+https://en.wikipedia.org/wiki/Catalan_number
+'''
+'''
+Our Contribution:
+Basically we Create the 2 function:
+    1. CountBST(n node)
+        CountBST Function Count the Number of Binary Search Tree Crated By n number of Node.
+        INPUT: N number i.e Number of Nodes
+        OUTPUT: Count i.e Number of Possible BST
+    1. CountBT(n node)
+        CountBT Function Count the Number of Binary Tree Crated By n number of Node.
+        INPUT: N number i.e Number of Nodes
+        OUTPUT: Count i.e Number of Possible BT
+        
+    >> CountBST(5)
+    >>  42
+    >> CountBST(6)
+    >> 132
+    >> CountBT(5)
+    >> 5040
+    >> CountBT(6)
+    >> 95040
+    
+    
+    >>> For Count of Binary Search Tree We find the Catalan Number Using Binomial Coefficient.
+    >>> For Count of Binary Tree We find the Factorial as well as Binomial Coefficient. Multiply both of them to get 
+        Number of Count of Binary Tree.
+'''
+
+def binomial_coefficient(n, k):
+    '''
+    Since Here we Find the Binomial Coefficient:
+    https://en.wikipedia.org/wiki/Binomial_coefficient
+    C(n,k) = n! / k!(n-k)!
+    :param n: 2 times of Number of Nodes
+    :param k: Number of Nodes
+    :return:  Integer Value
+    '''
+    result = 1    # To kept the Calculated Value
+    # Since C(n, k) = C(n, n-k)
+    if k > (n - k):
+        k = n - k
+    #Calculate C(n,k)
+    for i in range(k):
+        result *= (n - i)   #result = result * (n - i)
+        result //= (i + 1)  # result = result // (i + 1)
+
+    return result
+
+'''
+We can find Catalan Number in many Ways:
+But here we use Binomial Coefficent because it done the Job in O(n)
+'''
+def catalan(n):
+    '''
+    This Function will Find the Nth Catalan Number.
+    :param n: n number of Nodes
+    :return: Nth catalan Number
+    '''
+    value = binomial_coefficient(2 * n, n);
+    # We have to Find the Catalan Number Using 2nCn/(n+1)
+    catalan_number = value // (n + 1)
+    return catalan_number   # return nth catalan Number
+
+def CountBST(n):
+    '''
+    This Function Count the Number of Possible Binary Search Tree from Given Number of Nodes.
+    :param n: Number of Nodes
+    :return: Number : Total Number of Binary Search Tree Formed By Nodes
+    '''
+    count = catalan(n)  #find the Catalan Number at N index
+    return count    #Return the Number of Count
+
+def factorial(n):
+    '''
+    This Fuction use to Calculate The Factorial of N Number.
+    :param n: Nth Value to find the Factorial
+    :return: Factorial of Nth Value.
+    '''
+    result = 1  #kept the Result
+    #Calculate the Value upto 1 * 2... n
+    for i in range(1, n+1):
+        result *= i # result = result * i
+    return result
+
+def CountBT(n):
+    '''
+    This Function return the Count of Binary Tree.
+    :param n: n Number of Node
+    :return: Count of Binary Tree
+    '''
+    catalan_value = catalan(n)  #find the nth Catalan Number
+    factorial_value = factorial(n)  #find the factorial Value
+    count = catalan_value * factorial_value #find the Count of Binary Tree
+    return count  #return the Count
+if __name__ == '__main__':
+    n = int(input('Enter the Number of Node: '))
+    print('Total Number of Possible Binary Search Tree from ',n, ' Number of node ',CountBST(n))
+    print('Total Number of Possible Binary Search Tree from ',n, ' Number of node ' ,CountBT(n))
+
+