code: 
  
  
  class 						snake_case			:
  			'''simple docstring'''
  
  			def __init__(	self     :     Optional[Any]							,							lowerCAmelCase_     :     int					)					->					Optional[Any]:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       val
  								# Number of nodes in left subtree
  								SCREAMING_SNAKE_CASE_	      =       0
  								SCREAMING_SNAKE_CASE_	      =       None
  								SCREAMING_SNAKE_CASE_	      =       None
  								SCREAMING_SNAKE_CASE_	      =       None
  
  
  
  
  			def _lowercase (	self     :     str							,							lowerCAmelCase_     :     str					)					->					int:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"
  
  								if self.val < other.val:
  													SCREAMING_SNAKE_CASE_	      =       self.right
  													SCREAMING_SNAKE_CASE_	      =       None
  													if self.right:
  																		SCREAMING_SNAKE_CASE_	      =       other
  													SCREAMING_SNAKE_CASE_	      =       other
  													SCREAMING_SNAKE_CASE_	      =       self.left_tree_size * 2 + 1
  													return self
  								else:
  													SCREAMING_SNAKE_CASE_	      =       other.right
  													SCREAMING_SNAKE_CASE_	      =       None
  													if other.right:
  																		SCREAMING_SNAKE_CASE_	      =       self
  													SCREAMING_SNAKE_CASE_	      =       self
  													SCREAMING_SNAKE_CASE_	      =       other.left_tree_size * 2 + 1
  													return other
  
  class 						snake_case			:
  			'''simple docstring'''
  
  			def __init__(	self     :     Any							,							lowerCAmelCase_     :     Dict=None							,							lowerCAmelCase_     :     List[str]=None							,							lowerCAmelCase_     :     List[str]=0					)					->					List[Any]:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       heap_size
  								SCREAMING_SNAKE_CASE_	      =       bottom_root
  								SCREAMING_SNAKE_CASE_	      =       min_node
  
  			def _lowercase (	self     :     Optional[Any]							,							lowerCAmelCase_     :     Dict					)					->					str:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if other.size == 0:
  													return None
  								if self.size == 0:
  													SCREAMING_SNAKE_CASE_	      =       other.size
  													SCREAMING_SNAKE_CASE_	      =       other.bottom_root
  													SCREAMING_SNAKE_CASE_	      =       other.min_node
  													return None
  								# Update size
  								SCREAMING_SNAKE_CASE_	      =       self.size + other.size
  
  								# Update min.node
  								if self.min_node.val > other.min_node.val:
  													SCREAMING_SNAKE_CASE_	      =       other.min_node
  								# Merge
  
  								# Order roots by left_subtree_size
  								SCREAMING_SNAKE_CASE_	      =       []
  								SCREAMING_SNAKE_CASE_      ,SCREAMING_SNAKE_CASE_	      =       self.bottom_root, other.bottom_root
  								while i or j:
  													if i and ((not j) or i.left_tree_size < j.left_tree_size):
  																		combined_roots_list.append((i, True)					)
  																		SCREAMING_SNAKE_CASE_	      =       i.parent
  													else:
  																		combined_roots_list.append((j, False)					)
  																		SCREAMING_SNAKE_CASE_	      =       j.parent
          # Insert links between them
  								for i in range(len(lowerCAmelCase_					) - 1					):
  													if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
  																		SCREAMING_SNAKE_CASE_	      =       combined_roots_list[i + 1][0]
  																		SCREAMING_SNAKE_CASE_	      =       combined_roots_list[i][0]
          # Consecutively merge roots with same left_tree_size
  								SCREAMING_SNAKE_CASE_	      =       combined_roots_list[0][0]
  								while i.parent:
  													if (
  													    (i.left_tree_size == i.parent.left_tree_size) and (not i.parent.parent)
  													) or (
  													    i.left_tree_size == i.parent.left_tree_size
  													    and i.left_tree_size != i.parent.parent.left_tree_size
  													):
  																		# Neighbouring Nodes
  																		SCREAMING_SNAKE_CASE_	      =       i.left
  																		SCREAMING_SNAKE_CASE_	      =       i.parent.parent
  
  																		# Merging trees
  																		SCREAMING_SNAKE_CASE_	      =       i.merge_trees(i.parent					)
  
  																		# Updating links
  																		SCREAMING_SNAKE_CASE_	      =       previous_node
  																		SCREAMING_SNAKE_CASE_	      =       next_node
  																		if previous_node:
  																							SCREAMING_SNAKE_CASE_	      =       i
  																		if next_node:
  																							SCREAMING_SNAKE_CASE_	      =       i
  													else:
  																		SCREAMING_SNAKE_CASE_	      =       i.parent
          # Updating self.bottom_root
  								while i.left:
  													SCREAMING_SNAKE_CASE_	      =       i.left
  								SCREAMING_SNAKE_CASE_	      =       i
  
  								# Update other
  								SCREAMING_SNAKE_CASE_	      =       self.size
  								SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  								SCREAMING_SNAKE_CASE_	      =       self.min_node
  
  								# Return the merged heap
  								return self
  
  			def _lowercase (	self     :     Optional[int]							,							lowerCAmelCase_     :     List[Any]					)					->					Optional[Any]:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if self.size == 0:
  													SCREAMING_SNAKE_CASE_	      =       Node(lowerCAmelCase_					)
  													SCREAMING_SNAKE_CASE_	      =       1
  													SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  								else:
  													# Create new node
  													SCREAMING_SNAKE_CASE_	      =       Node(lowerCAmelCase_					)
  
  													# Update size
  													self.size += 1
  
  													# update min_node
  													if val < self.min_node.val:
  																		SCREAMING_SNAKE_CASE_	      =       new_node
  													# Put new_node as a bottom_root in heap
  													SCREAMING_SNAKE_CASE_	      =       new_node
  													SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  													SCREAMING_SNAKE_CASE_	      =       new_node
  
  													# Consecutively merge roots with same left_tree_size
  													while (
  													    self.bottom_root.parent
  													    and self.bottom_root.left_tree_size
  													    == self.bottom_root.parent.left_tree_size
  													):
  																		# Next node
  																		SCREAMING_SNAKE_CASE_	      =       self.bottom_root.parent.parent
  
  																		# Merge
  																		SCREAMING_SNAKE_CASE_	      =       self.bottom_root.merge_trees(self.bottom_root.parent					)
  
  																		# Update Links
  																		SCREAMING_SNAKE_CASE_	      =       next_node
  																		SCREAMING_SNAKE_CASE_	      =       None
  																		if next_node:
  																							SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  
  			def _lowercase (	self     :     Dict					)					->					int:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								return self.min_node.val
  
  			def _lowercase (	self     :     List[Any]					)					->					List[str]:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								return self.size == 0
  
  			def _lowercase (	self     :     Optional[int]					)					->					Any:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       self.min_node.val
  
  								# Last element in heap corner case
  								if self.size == 1:
  													# Update size
  													SCREAMING_SNAKE_CASE_	      =       0
  
  													# Update bottom root
  													SCREAMING_SNAKE_CASE_	      =       None
  
  													# Update min_node
  													SCREAMING_SNAKE_CASE_	      =       None
  
  													return min_value
  								# No right subtree corner case
  								# The structure of the tree implies that this should be the bottom root
  								# and there is at least one other root
  								if self.min_node.right is None:
  													# Update size
  													self.size -= 1
  
  													# Update bottom root
  													SCREAMING_SNAKE_CASE_	      =       self.bottom_root.parent
  													SCREAMING_SNAKE_CASE_	      =       None
  
  													# Update min_node
  													SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  													SCREAMING_SNAKE_CASE_	      =       self.bottom_root.parent
  													while i:
  																		if i.val < self.min_node.val:
  																							SCREAMING_SNAKE_CASE_	      =       i
  																		SCREAMING_SNAKE_CASE_	      =       i.parent
  													return min_value
  								# General case
  								# Find the BinomialHeap of the right subtree of min_node
  								SCREAMING_SNAKE_CASE_	      =       self.min_node.right
  								SCREAMING_SNAKE_CASE_	      =       None
  								SCREAMING_SNAKE_CASE_	      =       bottom_of_new
  								SCREAMING_SNAKE_CASE_	      =       1
  
  								# Size, min_node and bottom_root
  								while bottom_of_new.left:
  													SCREAMING_SNAKE_CASE_	      =       size_of_new * 2 + 1
  													SCREAMING_SNAKE_CASE_	      =       bottom_of_new.left
  													if bottom_of_new.val < min_of_new.val:
  																		SCREAMING_SNAKE_CASE_	      =       bottom_of_new
          # Corner case of single root on top left path
  								if (not self.min_node.left) and (not self.min_node.parent):
  													SCREAMING_SNAKE_CASE_	      =       size_of_new
  													SCREAMING_SNAKE_CASE_	      =       bottom_of_new
  													SCREAMING_SNAKE_CASE_	      =       min_of_new
  													# print("Single root, multiple nodes case")
  													return min_value
  								# Remaining cases
  								# Construct heap of right subtree
  								SCREAMING_SNAKE_CASE_	      =       BinomialHeap(
  								    bottom_root=lowerCAmelCase_							,							min_node=lowerCAmelCase_							,							heap_size=lowerCAmelCase_					)
  
  								# Update size
  								SCREAMING_SNAKE_CASE_	      =       self.size - 1 - size_of_new
  
  								# Neighbour nodes
  								SCREAMING_SNAKE_CASE_	      =       self.min_node.left
  								SCREAMING_SNAKE_CASE_	      =       self.min_node.parent
  
  								# Initialize new bottom_root and min_node
  								SCREAMING_SNAKE_CASE_	      =       previous_node or next_node
  								SCREAMING_SNAKE_CASE_	      =       next_node
  
  								# Update links of previous_node and search below for new min_node and
  								# bottom_root
  								if previous_node:
  													SCREAMING_SNAKE_CASE_	      =       next_node
  
  													# Update bottom_root and search for min_node below
  													SCREAMING_SNAKE_CASE_	      =       previous_node
  													SCREAMING_SNAKE_CASE_	      =       previous_node
  													while self.bottom_root.left:
  																		SCREAMING_SNAKE_CASE_	      =       self.bottom_root.left
  																		if self.bottom_root.val < self.min_node.val:
  																							SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  								if next_node:
  													SCREAMING_SNAKE_CASE_	      =       previous_node
  
  													# Search for new min_node above min_node
  													SCREAMING_SNAKE_CASE_	      =       next_node
  													while i:
  																		if i.val < self.min_node.val:
  																							SCREAMING_SNAKE_CASE_	      =       i
  																		SCREAMING_SNAKE_CASE_	      =       i.parent
          # Merge heaps
  								self.merge_heaps(lowerCAmelCase_					)
  
  								return min_value
  
  			def _lowercase (	self     :     Tuple					)					->					Tuple:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       self.bottom_root
  								while top_root.parent:
  													SCREAMING_SNAKE_CASE_	      =       top_root.parent
  								# preorder
  								SCREAMING_SNAKE_CASE_	      =       []
  								self.__traversal(lowerCAmelCase_							,							lowerCAmelCase_					)
  								return heap_pre_order
  
  			def _lowercase (	self     :     List[Any]							,							lowerCAmelCase_     :     List[str]							,							lowerCAmelCase_     :     int							,							lowerCAmelCase_     :     Union[str, Any]=0					)					->					str:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if curr_node:
  													preorder.append((curr_node.val, level)					)
  													self.__traversal(curr_node.left							,							lowerCAmelCase_							,							level + 1					)
  													self.__traversal(curr_node.right							,							lowerCAmelCase_							,							level + 1					)
  								else:
  													preorder.append(('''#''', level)					)
  
  
  
  
  			def __str__(	self     :     str					)					->					Union[str, Any]:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if self.is_empty():
  													return ""
  								SCREAMING_SNAKE_CASE_	      =       self.pre_order()
  
  								return "\n".join(('''-''' * level + str(lowerCAmelCase_					)) for value, level in preorder_heap					)
  
  
  # Unit Tests
  if __name__ == "__main__":
  		import doctest
  
  		doctest.testmod()
  
  
  
style_context: 
  
  
  from __future__ import annotations
  
  class 						snake_case			:
  			'''simple docstring'''
  
  			def __init__(	self     :     int							,							lowerCAmelCase_     :     int					)					->					None:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       value
  								SCREAMING_SNAKE_CASE_	      =       None
  								SCREAMING_SNAKE_CASE_	      =       None
  
  class 						snake_case			:
  			'''simple docstring'''
  
  			UpperCAmelCase       :       int
  
  			def __init__(	self     :     Union[str, Any]					)					->					None:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								SCREAMING_SNAKE_CASE_	      =       0
  
  			def _lowercase (	self     :     Optional[int]							,							lowerCAmelCase_     :     Node | None							,							lowerCAmelCase_     :     int					)					->					None:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if node is None:
  													return
  
  								if path_sum == self.target:
  													self.paths += 1
  
  								if node.left:
  													self.depth_first_search(node.left							,							path_sum + node.left.value					)
  								if node.right:
  													self.depth_first_search(node.right							,							path_sum + node.right.value					)
  
  
  
  
  			def _lowercase (	self     :     str							,							lowerCAmelCase_     :     Node | None							,							lowerCAmelCase_     :     int | None = None					)					->					int:
  
  
  
  
  								"""simple docstring"""
  
  
  
  
  
  
  								if node is None:
  													return 0
  								if target is not None:
  													SCREAMING_SNAKE_CASE_	      =       target
  
  								self.depth_first_search(lowerCAmelCase_							,							node.value					)
  								self.path_sum(node.left					)
  								self.path_sum(node.right					)
  
  								return self.paths
  
  
  if __name__ == "__main__":
  		import doctest
  
  		doctest.testmod()
  
  
  