data_structures.binary_tree.number_of_possible_binary_trees
===========================================================

.. py:module:: data_structures.binary_tree.number_of_possible_binary_trees

.. autoapi-nested-parse::

   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



Attributes
----------

.. autoapisummary::

   data_structures.binary_tree.number_of_possible_binary_trees.node_count


Functions
---------

.. autoapisummary::

   data_structures.binary_tree.number_of_possible_binary_trees.binary_tree_count
   data_structures.binary_tree.number_of_possible_binary_trees.binomial_coefficient
   data_structures.binary_tree.number_of_possible_binary_trees.catalan_number
   data_structures.binary_tree.number_of_possible_binary_trees.factorial


Module Contents
---------------

.. py:function:: binary_tree_count(node_count: int) -> int

   Return the number of possible of binary trees.
   :param n: number of nodes
   :return: Number of possible binary trees

   >>> binary_tree_count(5)
   5040
   >>> binary_tree_count(6)
   95040


.. py:function:: 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


.. py:function:: catalan_number(node_count: int) -> int

   We can find Catalan number many ways but here we use Binomial Coefficient 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


.. py:function:: 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


.. py:data:: node_count