def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		SCREAMING_SNAKE_CASE_  :		List[str]				      =    generate_pascal_triangle(lowerCAmelCase	)
		for row_idx in range(lowerCAmelCase	):
				# Print left spaces
				for _ in range(num_rows - row_idx - 1	):
						print(end=" "	)
				# Print row values
				for col_idx in range(row_idx + 1	):
						if col_idx != row_idx:
								print(triangle[row_idx][col_idx]   , end=" "	)
						else:
								print(triangle[row_idx][col_idx]   , end=""	)
				print()




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		if not isinstance(lowerCAmelCase   , lowerCAmelCase	):
				raise TypeError("The input value of 'num_rows' should be 'int'"	)

		if num_rows == 0:
				return []
		elif num_rows < 0:
				raise ValueError(
				    "The input value of 'num_rows' should be greater than or equal to 0"	)

		SCREAMING_SNAKE_CASE_  :		list[list[int]]				      =    []
		for current_row_idx in range(lowerCAmelCase	):
				SCREAMING_SNAKE_CASE_  :		str				      =    populate_current_row(lowerCAmelCase   , lowerCAmelCase	)
				triangle.append(lowerCAmelCase	)
		return triangle




def       _snake_case	(   lowerCAmelCase							:   list[list[int]]   , lowerCAmelCase							:   int	):
		"""simple docstring"""







		SCREAMING_SNAKE_CASE_  :		str				      =    [-1] * (current_row_idx + 1)
		# first and last elements of current row are equal to 1
		SCREAMING_SNAKE_CASE_      ,    SCREAMING_SNAKE_CASE_  :		Tuple				      =    1, 1
		for current_col_idx in range(1   , lowerCAmelCase	):
				calculate_current_element(
				    lowerCAmelCase   , lowerCAmelCase   , lowerCAmelCase   , lowerCAmelCase	)
		return current_row




def       _snake_case	(   lowerCAmelCase							:   list[list[int]]   , lowerCAmelCase							:   list[int]   , lowerCAmelCase							:   int   , lowerCAmelCase							:   int   , ):
		"""simple docstring"""







		SCREAMING_SNAKE_CASE_  :		Any				      =    triangle[current_row_idx - 1][current_col_idx - 1]
		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    triangle[current_row_idx - 1][current_col_idx]
		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    above_to_left_elt + above_to_right_elt




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		if not isinstance(lowerCAmelCase   , lowerCAmelCase	):
				raise TypeError("The input value of 'num_rows' should be 'int'"	)

		if num_rows == 0:
				return []
		elif num_rows < 0:
				raise ValueError(
				    "The input value of 'num_rows' should be greater than or equal to 0"	)

		SCREAMING_SNAKE_CASE_  :		list[list[int]]				      =    [[1]]

		for row_index in range(1   , lowerCAmelCase	):
				SCREAMING_SNAKE_CASE_  :		Tuple				      =    [0] + result[-1] + [0]
				SCREAMING_SNAKE_CASE_  :		List[str]				      =    row_index + 1
				# Calculate the number of distinct elements in a row
				SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    sum(divmod(lowerCAmelCase   , 2	)	)
				SCREAMING_SNAKE_CASE_  :		List[Any]				      =    [
				    temp_row[i - 1] + temp_row[i] for i in range(1   , distinct_elements + 1	)
				]
				SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    row_first_half[: (row_index + 1) // 2]
				row_second_half.reverse()
				SCREAMING_SNAKE_CASE_  :		Tuple				      =    row_first_half + row_second_half
				result.append(lowerCAmelCase	)

		return result




def       _snake_case	(   ):
		"""simple docstring"""







		from collections.abc import Callable
		from timeit import timeit

		def benchmark_a_function(lowerCAmelCase							:   Callable   , lowerCAmelCase							:   int	) -> None:
				SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    f'{func.__name__}({value})'
				SCREAMING_SNAKE_CASE_  :		str				      =    timeit(f'__main__.{call}'   , setup="import __main__"	)
				# print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
				print(f'{call:38} -- {timing:.4f} seconds'	)

		for value in range(1_5	):  # (1, 7, 14):
				for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
						benchmark_a_function(lowerCAmelCase   , lowerCAmelCase	)
				print()


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		benchmark()

from collections import defaultdict




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		SCREAMING_SNAKE_CASE_  :		Any				      =    1
		SCREAMING_SNAKE_CASE_  :		Tuple				      =    True
		for v in tree[start]:
				if v not in visited:
						ret += dfs(lowerCAmelCase	)
		if ret % 2 == 0:
				cuts.append(lowerCAmelCase	)
		return ret




def       _snake_case	(   ):
		"""simple docstring"""







		dfs(1	)


if __name__ == "__main__":
		__lowerCamelCase   ,						__lowerCamelCase					:	Union[str, Any]       =  10, 9
		__lowerCamelCase					:	Optional[int]       =  defaultdict(list)
		__lowerCamelCase					:	dict[int, bool]       =  {}
		__lowerCamelCase					:	list[int]       =  []
		__lowerCamelCase					:	Optional[Any]       =  0
		__lowerCamelCase					:	Any       =  [(2, 1), (3, 1), (4, 3), (5, 2), (6, 1), (7, 2), (8, 6), (9, 8), (10, 8)]
		for u, v in edges:
				tree[u].append(v)
				tree[v].append(u)
		even_tree()
		print(len(cuts) - 1)
