New Code!!(Finding the N Possible Binary Search Tree and Binary Tree from Given N node Number) (#1663)

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* Code Upload

* Delete n_possible_bst

* Find the N Possible Binary Tree and Binary Tree from given Nth Number of Node.

* Update in Test

* Update and rename n_possible_bst.py to number_of_possible_binary_trees.py

Co-authored-by: Christian Clauss <cclauss@me.com>

data_structures/binary_tree/number_of_possible_binary_trees.py

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+"""
+Hey, we are going to find an exciting number called Catalan number which is use to find
+the number of possible binary search trees from tree of a given number of nodes.
+
+We will use the formula: t(n) = SUMMATION(i = 1 to n)t(i-1)t(n-i)
+
+Further details at Wikipedia: https://en.wikipedia.org/wiki/Catalan_number
+"""
+"""
+Our Contribution:
+Basically we Create the 2 function:
+    1. catalan_number(node_count: int) -> int
+        Returns the number of possible binary search trees for n nodes.
+    2. binary_tree_count(node_count: int) -> int
+        Returns the number of possible binary trees for n nodes.
+"""
+
+
+def binomial_coefficient(n: int, k: int) -> int:
+    """
+    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
+
+    >>> binomial_coefficient(4, 2)
+    6
+    """
+    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 //= i + 1
+    return result
+
+
+def catalan_number(node_count: int) -> int:
+    """
+    We can find Catalan number many ways but here we use Binomial Coefficent because it
+    does the job in O(n)
+
+    return the Catalan number of n using 2nCn/(n+1).
+    :param n: number of nodes
+    :return: Catalan number of n nodes
+
+    >>> catalan_number(5)
+    42
+    >>> catalan_number(6)
+    132
+    """
+    return binomial_coefficient(2 * node_count, node_count) // (node_count + 1)
+
+
+def factorial(n: int) -> int:
+    """
+    Return the factorial of a number.
+    :param n: Number to find the Factorial of.
+    :return: Factorial of n.
+
+    >>> import math
+    >>> all(factorial(i) == math.factorial(i) for i in range(10))
+    True
+    >>> factorial(-5)  # doctest: +ELLIPSIS
+    Traceback (most recent call last):
+    ...
+    ValueError: factorial() not defined for negative values
+    """
+    if n < 0:
+        raise ValueError("factorial() not defined for negative values")
+    result = 1
+    for i in range(1, n + 1):
+        result *= i
+    return result
+
+
+def binary_tree_count(node_count: int) -> int:
+    """
+    Return the number of possible of binary trees.
+    :param n: number of nodes
+    :return: Number of possilble binary trees
+
+    >>> binary_tree_count(5)
+    5040
+    >>> binary_tree_count(6)
+    95040
+    """
+    return catalan_number(node_count) * factorial(node_count)
+
+
+if __name__ == "__main__":
+    node_count = int(input("Enter the number of nodes: ").strip() or 0)
+    if node_count <= 0:
+        raise ValueError("We need some nodes to work with.")
+    print(
+        f"Given {node_count} nodes, there are {binary_tree_count(node_count)} "
+        f"binary trees and {catalan_number(node_count)} binary search trees."
+    )