




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




def 		lowerCAmelCase_ (						lowerCamelCase ):
				return binomial_coefficient(2 * node_count						,  lowerCamelCase ) // (node_count + 1)




def 		lowerCAmelCase_ (						lowerCamelCase ):
				if n < 0:
								raise ValueError("""factorial() not defined for negative values""" )
				__magic_name__						:						List[str]										=1
				for i in range(1						,  n + 1 ):
								result *= i
				return result




def 		lowerCAmelCase_ (						lowerCamelCase ):
				return catalan_number(lowerCamelCase ) * factorial(lowerCamelCase )


if __name__ == "__main__":
							UpperCAmelCase_		:					Union[str, Any]          =  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."""
							)










from __future__ import annotations

from fractions import Fraction




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase ):
				return (
				    num != den and num % 10 == den // 10 and (num // 10) / (den % 10) == num / den
				)




def 		lowerCAmelCase_ (						lowerCamelCase ):
				__magic_name__						:						List[str]										=[]
				__magic_name__						:						List[Any]										=11
				__magic_name__						:						Tuple										=int("""1""" + """0""" * digit_len )
				for num in range(lowerCamelCase						,  lowerCamelCase ):
								while den <= 99:
												if (num != den) and (num % 10 == den // 10) and (den % 10 != 0):
																if is_digit_cancelling(lowerCamelCase						,  lowerCamelCase ):
																				solutions.append(F"{num}/{den}" )
												den += 1
								num += 1
								__magic_name__						:						List[str]										=10
				return solutions




def 		lowerCAmelCase_ (						lowerCamelCase = 2 ):
				__magic_name__						:						str										=1.0
				for fraction in fraction_list(lowerCamelCase ):
								__magic_name__						:						int										=Fraction(lowerCamelCase )
								result *= frac.denominator / frac.numerator
				return int(lowerCamelCase )


if __name__ == "__main__":
							print(solution())




