
'''simple docstring'''







def 					snake_case_		(				__SCREAMING_SNAKE_CASE					: int      ,      __SCREAMING_SNAKE_CASE					: int							):



						"""simple docstring"""





						lowercase_	:     str													=       1  # To kept the Calculated Value
						# Since C(n, k) = C(n, n-k)
						if k > (n - k):
												lowercase_	:     Any													=       n - k
						# Calculate C(n,k)
						for i in range(__SCREAMING_SNAKE_CASE							):
												result *= n - i
												result //= i + 1
						return result
def 					snake_case_		(				__SCREAMING_SNAKE_CASE					: int							):



						"""simple docstring"""





						return binomial_coefficient(2 * node_count      ,      __SCREAMING_SNAKE_CASE							) // (node_count + 1)
def 					snake_case_		(				__SCREAMING_SNAKE_CASE					: int							):



						"""simple docstring"""





						if n < 0:
												raise ValueError('''factorial() not defined for negative values'''							)
						lowercase_	:     List[Any]													=       1
						for i in range(1      ,      n + 1							):
												result *= i
						return result
def 					snake_case_		(				__SCREAMING_SNAKE_CASE					: int							):



						"""simple docstring"""





						return catalan_number(__SCREAMING_SNAKE_CASE							) * factorial(__SCREAMING_SNAKE_CASE							)


if __name__ == "__main__":
							_lowercase :  Tuple    							=       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

from collections import Counter
from random import random






class  lowerCAmelCase__   :






					def __init__(   self		):


											"""simple docstring"""





											lowercase_	:     int													=       {}






					def 	_snake_case    (   self				,							__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											lowercase_	:     Dict													=       {}






					def 	_snake_case    (   self				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE				,							__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											if nodea not in self.connections:
																	self.add_node(__SCREAMING_SNAKE_CASE		)
											if nodea not in self.connections:
																	self.add_node(__SCREAMING_SNAKE_CASE		)
											lowercase_	:     Optional[Any]													=       probability






					def 	_snake_case    (   self		):


											"""simple docstring"""





											return list(self.connections		)






					def 	_snake_case    (   self				,							__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											lowercase_	:     Any													=       0
											lowercase_	:     Tuple													=       random()

											for dest in self.connections[node]:
																	current_probability += self.connections[node][dest]
																	if current_probability > random_value:
																							return dest
											return ""







def 					snake_case_		(				__SCREAMING_SNAKE_CASE					: str      ,      __SCREAMING_SNAKE_CASE					: list[tuple[str, str, float]]      ,      __SCREAMING_SNAKE_CASE					: int							):



						"""simple docstring"""





						lowercase_	:     List[Any]													=       MarkovChainGraphUndirectedUnweighted()

						for nodea, nodea, probability in transitions:
												graph.add_transition_probability(__SCREAMING_SNAKE_CASE      ,      __SCREAMING_SNAKE_CASE      ,      __SCREAMING_SNAKE_CASE							)

						lowercase_	:     str													=       Counter(graph.get_nodes()							)
						lowercase_	:     Any													=       start

						for _ in range(__SCREAMING_SNAKE_CASE							):
												lowercase_	:     int													=       graph.transition(__SCREAMING_SNAKE_CASE							)
												visited[node] += 1

						return visited


if __name__ == "__main__":
							import doctest

							doctest.testmod()



