





class 			lowercase				:



						"""simple docstring"""

						def __init__(    self			:		Union[str, Any]					):





										'''simple docstring'''







										_snake_case :			dict[str, TrieNode]  						=   {}  # Mapping from char to TrieNode
										_snake_case :			Dict  						=   False







						def 		__UpperCAmelCase       (    self			:		Dict ,				lowerCamelCase_			:		list[str]					):





										'''simple docstring'''







										for word in words:
														self.insert(lowerCamelCase_					)







						def 		__UpperCAmelCase       (    self			:		Optional[int] ,				lowerCamelCase_			:		str					):





										'''simple docstring'''







										_snake_case :			str  						=   self
										for char in word:
														if char not in curr.nodes:
																		_snake_case :			List[Any]  						=   TrieNode()
														_snake_case :			int  						=   curr.nodes[char]
										_snake_case :			Optional[int]  						=   True







						def 		__UpperCAmelCase       (    self			:		Union[str, Any] ,				lowerCamelCase_			:		str					):





										'''simple docstring'''







										_snake_case :			Tuple  						=   self
										for char in word:
														if char not in curr.nodes:
																		return False
														_snake_case :			Optional[int]  						=   curr.nodes[char]
										return curr.is_leaf







						def 		__UpperCAmelCase       (    self			:		Dict ,				lowerCamelCase_			:		str					):





										'''simple docstring'''







										def _delete(lowerCamelCase_			:		TrieNode ,				lowerCamelCase_			:		str ,				lowerCamelCase_			:		int					) -> bool:
														if index == len(lowerCamelCase_					):
																		# If word does not exist
																		if not curr.is_leaf:
																						return False
																		_snake_case :			Union[str, Any]  						=   False
																		return len(curr.nodes					) == 0
														_snake_case :			Dict  						=   word[index]
														_snake_case :			Tuple  						=   curr.nodes.get(lowerCamelCase_					)
														# If char not in current trie node
														if not char_node:
																		return False
														# Flag to check if node can be deleted
														_snake_case :			Dict  						=   _delete(lowerCamelCase_ ,				lowerCamelCase_ ,				index + 1					)
														if delete_curr:
																		del curr.nodes[char]
																		return len(curr.nodes					) == 0
														return delete_curr

										_delete(self ,				lowerCamelCase_ ,				0					)



def       A__(		__lowerCAmelCase	,			__lowerCAmelCase ):
				if node.is_leaf:
								print(__lowerCAmelCase	,			end=' ' )

				for key, value in node.nodes.items():
								print_words(__lowerCAmelCase	,			word + key )

def       A__(		):
				_snake_case :			Union[str, Any]  						=   'banana bananas bandana band apple all beast'.split()
				_snake_case :			Union[str, Any]  						=   TrieNode()
				root.insert_many(__lowerCAmelCase )
				# print_words(root, "")
				assert all(root.find(__lowerCAmelCase ) for word in words )
				assert root.find('banana' )
				assert not root.find('bandanas' )
				assert not root.find('apps' )
				assert root.find('apple' )
				assert root.find('all' )
				root.delete('all' )
				assert not root.find('all' )
				root.delete('banana' )
				assert not root.find('banana' )
				assert root.find('bananas' )
				return True

def       A__(		__lowerCAmelCase	,			__lowerCAmelCase ):
				print(str(__lowerCAmelCase )	,			'works!' if passes else 'doesn\'t work :(' )

def       A__(		):
				assert test_trie()

def       A__(		):
				print_results('Testing trie functionality'	,			test_trie() )


if __name__ == "__main__":
		main()













from math import factorial

def       A__(		__lowerCAmelCase	,			__lowerCAmelCase	,			__lowerCAmelCase ):
				if successes > trials:
								raise ValueError('successes must be lower or equal to trials' )
				if trials < 0 or successes < 0:
								raise ValueError('the function is defined for non-negative integers' )
				if not isinstance(__lowerCAmelCase	,			__lowerCAmelCase ) or not isinstance(__lowerCAmelCase	,			__lowerCAmelCase ):
								raise ValueError('the function is defined for non-negative integers' )
				if not 0 < prob < 1:
								raise ValueError('prob has to be in range of 1 - 0' )
				_snake_case :			Optional[int]  						=   (prob**successes) * ((1 - prob) ** (trials - successes))
				# Calculate the binomial coefficient: n! / k!(n-k)!
				_snake_case :			List[Any]  						=   float(factorial(__lowerCAmelCase ) )
				coefficient /= factorial(__lowerCAmelCase ) * factorial(trials - successes )
				return probability * coefficient


if __name__ == "__main__":
		from doctest import testmod

		testmod()
		print('''Probability of 2 successes out of 4 trails''')
		print('''with probability of 0.75 is:''', end=''' ''')
		print(binomial_distribution(2, 4, 0.75))






