
'''simple docstring'''

import sys
from random import randint






class 				A       :





	def __init__(   self     :    str ,							__magic_name__     :    str ,							__magic_name__     :    Dict							):


				"""simple docstring"""

				lowerCAmelCase__									=							key
				lowerCAmelCase__									=							freq







	def __str__(   self     :    List[Any]							):


				"""simple docstring"""

				return f"""Node(key={self.key}, freq={self.freq})"""







def 					A       ( UpperCamelCase_   :       Union[str, Any]  ,		UpperCamelCase_   :       List[Any]  ,		UpperCamelCase_   :       Any  ,		UpperCamelCase_   :       int  ,		UpperCamelCase_   :       Optional[int]  ,		UpperCamelCase_   :       str )   ->    List[str]:







			'''simple docstring'''



			if i > j or i < 0 or j > len(UpperCamelCase_ ) - 1:
						return

			lowerCAmelCase__									=							root[i][j]
			if parent == -1:  # root does not have a parent
						print(F"""{key[node]} is the root of the binary search tree.""" )
			elif is_left:
						print(F"""{key[node]} is the left child of key {parent}.""" )
			else:
						print(F"""{key[node]} is the right child of key {parent}.""" )

			print_binary_search_tree(UpperCamelCase_  ,		UpperCamelCase_  ,		UpperCamelCase_  ,		node - 1  ,		key[node]  ,		UpperCamelCase_ )
			print_binary_search_tree(UpperCamelCase_  ,		UpperCamelCase_  ,		node + 1  ,		UpperCamelCase_  ,		key[node]  ,		UpperCamelCase_ )







def 					A       ( UpperCamelCase_   :       Optional[int] )   ->    Dict:







			'''simple docstring'''



			nodes.sort(key=lambda UpperCamelCase_ : node.key )

			lowerCAmelCase__									=							len(UpperCamelCase_ )

			lowerCAmelCase__									=							[nodes[i].key for i in range(UpperCamelCase_ )]
			lowerCAmelCase__									=							[nodes[i].freq for i in range(UpperCamelCase_ )]

			# This 2D array stores the overall tree cost (which's as minimized as possible);
			# for a single key, cost is equal to frequency of the key.
			lowerCAmelCase__									=							[[freqs[i] if i == j else 0 for j in range(UpperCamelCase_ )] for i in range(UpperCamelCase_ )]
			# sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes
			# array
			lowerCAmelCase__									=							[[freqs[i] if i == j else 0 for j in range(UpperCamelCase_ )] for i in range(UpperCamelCase_ )]
			# stores tree roots that will be used later for constructing binary search tree
			lowerCAmelCase__									=							[[i if i == j else 0 for j in range(UpperCamelCase_ )] for i in range(UpperCamelCase_ )]

			for interval_length in range(2  ,		n + 1 ):
						for i in range(n - interval_length + 1 ):
									lowerCAmelCase__									=							i + interval_length - 1

									lowerCAmelCase__									=							sys.maxsize  # set the value to "infinity"
									lowerCAmelCase__									=							total[i][j - 1] + freqs[j]

									# Apply Knuth's optimization
									# Loop without optimization: for r in range(i, j + 1):
									for r in range(root[i][j - 1]  ,		root[i + 1][j] + 1 ):  # r is a temporal root
												lowerCAmelCase__									=							dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
												lowerCAmelCase__									=							dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
												lowerCAmelCase__									=							left + total[i][j] + right

												if dp[i][j] > cost:
															lowerCAmelCase__									=							cost
															lowerCAmelCase__									=							r

			print("Binary search tree nodes:" )
			for node in nodes:
						print(UpperCamelCase_ )

			print(F"""\nThe cost of optimal BST for given tree nodes is {dp[0][n - 1]}.""" )
			print_binary_search_tree(UpperCamelCase_  ,		UpperCamelCase_  ,		0  ,		n - 1  ,		-1  ,		UpperCamelCase_ )







def 					A       ( )   ->    Union[str, Any]:







			'''simple docstring'''



			lowerCAmelCase__									=							[Node(UpperCamelCase_  ,		randint(1  ,		50 ) ) for i in range(10  ,		0  ,		-1 )]
			find_optimal_binary_search_tree(UpperCamelCase_ )


if __name__ == "__main__":
						main()







'''simple docstring'''

def 					A       ( )   ->    str:







			'''simple docstring'''



			lowerCAmelCase__									=							1

			while True:
						if (
						    sorted(str(UpperCamelCase_ ) )
						    == sorted(str(2 * i ) )
						    == sorted(str(3 * i ) )
						    == sorted(str(4 * i ) )
						    == sorted(str(5 * i ) )
						    == sorted(str(6 * i ) )
						):
									return i

						i += 1


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





