






'''simple docstring'''







def 		_UpperCAmelCase   (							__A							:   Any						,		__A							:   Optional[int]      ):
				a_		:  Any											=      1  # To kept the Calculated Value
				# Since C(n, k) = C(n, n-k)
				if k > (n - k):
								a_		:  List[str]											=      n - k
				# Calculate C(n,k)
				for i in range(__A      ):
								result *= n - i
								result //= i + 1
				return result





def 		_UpperCAmelCase   (							__A							:   Union[str, Any]      ):
				return binomial_coefficient(2 * node_count						,		__A      ) // (node_count + 1)





def 		_UpperCAmelCase   (							__A							:   List[str]      ):
				if n < 0:
								raise ValueError('''factorial() not defined for negative values'''      )
				a_		:  str											=      1
				for i in range(1						,		n + 1      ):
								result *= i
				return result





def 		_UpperCAmelCase   (							__A							:   Dict      ):
				return catalan_number(__A      ) * factorial(__A      )


if __name__ == "__main__":
		__lowerCAmelCase       		=      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."""
		)












'''simple docstring'''







from __future__ import annotations





def 		_UpperCAmelCase   (							__A							:   tuple[int, int]						,		__A							:   int      ):
				a_				,			a_		:  List[str]											=      position
				a_		:  Optional[int]											=      [
				    (y + 1, x + 2),
				    (y - 1, x + 2),
				    (y + 1, x - 2),
				    (y - 1, x - 2),
				    (y + 2, x + 1),
				    (y + 2, x - 1),
				    (y - 2, x + 1),
				    (y - 2, x - 1),
				]
				a_		:  Any											=      []

				for position in positions:
								a_				,			a_		:  Dict											=      position
								if 0 <= y_test < n and 0 <= x_test < n:
												permissible_positions.append(__A      )

				return permissible_positions





def 		_UpperCAmelCase   (							__A							:   list[list[int]]      ):
				return not any(elem == 0 for row in board for elem in row      )





def 		_UpperCAmelCase   (							__A							:   list[list[int]]						,		__A							:   tuple[int, int]						,		__A							:   int      ):

				if is_complete(__A      ):
								return True

				for position in get_valid_pos(__A						,		len(__A      )      ):
								a_				,			a_		:  Dict											=      position

								if board[y][x] == 0:
												a_		:  Optional[Any]											=      curr + 1
												if open_knight_tour_helper(__A						,		__A						,		curr + 1      ):
																return True
												a_		:  Tuple											=      0

				return False





def 		_UpperCAmelCase   (							__A							:   int      ):
				a_		:  List[str]											=      [[0 for i in range(__A      )] for j in range(__A      )]

				for i in range(__A      ):
								for j in range(__A      ):
												a_		:  Optional[Any]											=      1
												if open_knight_tour_helper(__A						,		(i, j)						,		1      ):
																return board
												a_		:  Union[str, Any]											=      0

				a_		:  Dict											=      f'Open Kight Tour cannot be performed on a board of size {n}'
				raise ValueError(__A      )


if __name__ == "__main__":
		import doctest

		doctest.testmod()




