

"""simple docstring"""
class        _UpperCamelCase	:


  """simple docstring"""





  def __init__( self					: List[str]       ,       snake_case					: Any      )	->			Tuple:


         '''simple docstring'''





         __magic_name__			:		Any        =     val
         # Number of nodes in left subtree
         __magic_name__			:		List[Any]        =     0
         __magic_name__			:		Optional[Any]        =     None
         __magic_name__			:		Tuple        =     None
         __magic_name__			:		Tuple        =     None




  def 							_UpperCAmelCase      ( self					: Optional[int]       ,       snake_case					: Tuple      )	->			Union[str, Any]:


         '''simple docstring'''





         assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

         if self.val < other.val:
                __magic_name__			:		Any        =     self.right
                __magic_name__			:		Union[str, Any]        =     None
                if self.right:
                       __magic_name__			:		List[str]        =     other
                __magic_name__			:		Dict        =     other
                __magic_name__			:		Dict        =     self.left_tree_size * 2 + 1
                return self
         else:
                __magic_name__			:		Optional[Any]        =     other.right
                __magic_name__			:		Dict        =     None
                if other.right:
                       __magic_name__			:		int        =     self
                __magic_name__			:		Optional[int]        =     self
                __magic_name__			:		List[str]        =     other.left_tree_size * 2 + 1
                return other

class        _UpperCamelCase	:


  """simple docstring"""





  def __init__( self					: int       ,       snake_case					: List[str]=None       ,       snake_case					: List[str]=None       ,       snake_case					: Optional[int]=0      )	->			Dict:


         '''simple docstring'''





         __magic_name__			:		str        =     heap_size
         __magic_name__			:		Optional[Any]        =     bottom_root
         __magic_name__			:		List[str]        =     min_node



  def 							_UpperCAmelCase      ( self					: List[Any]       ,       snake_case					: List[Any]      )	->			Optional[Any]:


         '''simple docstring'''





         if other.size == 0:
                return None
         if self.size == 0:
                __magic_name__			:		str        =     other.size
                __magic_name__			:		Union[str, Any]        =     other.bottom_root
                __magic_name__			:		Union[str, Any]        =     other.min_node
                return None
         # Update size
         __magic_name__			:		List[Any]        =     self.size + other.size

         # Update min.node
         if self.min_node.val > other.min_node.val:
                __magic_name__			:		Tuple        =     other.min_node
         # Merge

         # Order roots by left_subtree_size
         __magic_name__			:		Union[str, Any]        =     []
         __magic_name__       ,							__magic_name__			:		Dict        =     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)      )
                       __magic_name__			:		Union[str, Any]        =     i.parent
                else:
                       combined_roots_list.append((j, False)      )
                       __magic_name__			:		Dict        =     j.parent
        # Insert links between them
         for i in range(len(lowercase_      ) - 1      ):
                if combined_roots_list[i][1] != combined_roots_list[i + 1][1]:
                       __magic_name__			:		Optional[Any]        =     combined_roots_list[i + 1][0]
                       __magic_name__			:		str        =     combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
         __magic_name__			:		List[Any]        =     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
                       __magic_name__			:		Optional[int]        =     i.left
                       __magic_name__			:		List[str]        =     i.parent.parent

                       # Merging trees
                       __magic_name__			:		List[str]        =     i.merge_trees(i.parent      )

                       # Updating links
                       __magic_name__			:		List[Any]        =     previous_node
                       __magic_name__			:		List[Any]        =     next_node
                       if previous_node:
                              __magic_name__			:		Dict        =     i
                       if next_node:
                              __magic_name__			:		Optional[Any]        =     i
                else:
                       __magic_name__			:		Dict        =     i.parent
        # Updating self.bottom_root
         while i.left:
                __magic_name__			:		Optional[Any]        =     i.left
         __magic_name__			:		List[str]        =     i

         # Update other
         __magic_name__			:		Optional[Any]        =     self.size
         __magic_name__			:		int        =     self.bottom_root
         __magic_name__			:		List[Any]        =     self.min_node

         # Return the merged heap
         return self



  def 							_UpperCAmelCase      ( self					: int       ,       snake_case					: Union[str, Any]      )	->			Optional[int]:


         '''simple docstring'''





         if self.size == 0:
                __magic_name__			:		List[Any]        =     Node(lowercase_      )
                __magic_name__			:		str        =     1
                __magic_name__			:		List[Any]        =     self.bottom_root
         else:
                # Create new node
                __magic_name__			:		Any        =     Node(lowercase_      )

                # Update size
                self.size += 1

                # update min_node
                if val < self.min_node.val:
                       __magic_name__			:		List[Any]        =     new_node
                # Put new_node as a bottom_root in heap
                __magic_name__			:		List[str]        =     new_node
                __magic_name__			:		int        =     self.bottom_root
                __magic_name__			:		Any        =     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
                       __magic_name__			:		List[str]        =     self.bottom_root.parent.parent

                       # Merge
                       __magic_name__			:		Tuple        =     self.bottom_root.merge_trees(self.bottom_root.parent      )

                       # Update Links
                       __magic_name__			:		int        =     next_node
                       __magic_name__			:		Dict        =     None
                       if next_node:
                              __magic_name__			:		Tuple        =     self.bottom_root



  def 							_UpperCAmelCase      ( self					: List[Any]      )	->			Optional[Any]:


         '''simple docstring'''





         return self.min_node.val



  def 							_UpperCAmelCase      ( self					: List[str]      )	->			Any:


         '''simple docstring'''





         return self.size == 0



  def 							_UpperCAmelCase      ( self					: str      )	->			Any:


         '''simple docstring'''





         __magic_name__			:		Dict        =     self.min_node.val

         # Last element in heap corner case
         if self.size == 1:
                # Update size
                __magic_name__			:		Optional[int]        =     0

                # Update bottom root
                __magic_name__			:		Any        =     None

                # Update min_node
                __magic_name__			:		Optional[Any]        =     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
                __magic_name__			:		Optional[Any]        =     self.bottom_root.parent
                __magic_name__			:		str        =     None

                # Update min_node
                __magic_name__			:		List[Any]        =     self.bottom_root
                __magic_name__			:		Union[str, Any]        =     self.bottom_root.parent
                while i:
                       if i.val < self.min_node.val:
                              __magic_name__			:		List[Any]        =     i
                       __magic_name__			:		str        =     i.parent
                return min_value
         # General case
         # Find the BinomialHeap of the right subtree of min_node
         __magic_name__			:		List[str]        =     self.min_node.right
         __magic_name__			:		Tuple        =     None
         __magic_name__			:		str        =     bottom_of_new
         __magic_name__			:		int        =     1

         # Size, min_node and bottom_root
         while bottom_of_new.left:
                __magic_name__			:		Any        =     size_of_new * 2 + 1
                __magic_name__			:		Any        =     bottom_of_new.left
                if bottom_of_new.val < min_of_new.val:
                       __magic_name__			:		int        =     bottom_of_new
        # Corner case of single root on top left path
         if (not self.min_node.left) and (not self.min_node.parent):
                __magic_name__			:		int        =     size_of_new
                __magic_name__			:		Optional[Any]        =     bottom_of_new
                __magic_name__			:		List[str]        =     min_of_new
                # print("Single root, multiple nodes case")
                return min_value
         # Remaining cases
         # Construct heap of right subtree
         __magic_name__			:		int        =     BinomialHeap(
             bottom_root=lowercase_       ,       min_node=lowercase_       ,       heap_size=lowercase_      )

         # Update size
         __magic_name__			:		Optional[int]        =     self.size - 1 - size_of_new

         # Neighbour nodes
         __magic_name__			:		Any        =     self.min_node.left
         __magic_name__			:		Union[str, Any]        =     self.min_node.parent

         # Initialize new bottom_root and min_node
         __magic_name__			:		List[Any]        =     previous_node or next_node
         __magic_name__			:		List[str]        =     next_node

         # Update links of previous_node and search below for new min_node and
         # bottom_root
         if previous_node:
                __magic_name__			:		Union[str, Any]        =     next_node

                # Update bottom_root and search for min_node below
                __magic_name__			:		Union[str, Any]        =     previous_node
                __magic_name__			:		List[str]        =     previous_node
                while self.bottom_root.left:
                       __magic_name__			:		Tuple        =     self.bottom_root.left
                       if self.bottom_root.val < self.min_node.val:
                              __magic_name__			:		int        =     self.bottom_root
         if next_node:
                __magic_name__			:		Dict        =     previous_node

                # Search for new min_node above min_node
                __magic_name__			:		str        =     next_node
                while i:
                       if i.val < self.min_node.val:
                              __magic_name__			:		List[Any]        =     i
                       __magic_name__			:		Dict        =     i.parent
        # Merge heaps
         self.merge_heaps(lowercase_      )

         return min_value



  def 							_UpperCAmelCase      ( self					: List[str]      )	->			int:


         '''simple docstring'''





         __magic_name__			:		str        =     self.bottom_root
         while top_root.parent:
                __magic_name__			:		Dict        =     top_root.parent
         # preorder
         __magic_name__			:		Union[str, Any]        =     []
         self.__traversal(lowercase_       ,       lowercase_      )
         return heap_pre_order



  def 							_UpperCAmelCase      ( self					: str       ,       snake_case					: Dict       ,       snake_case					: Dict       ,       snake_case					: List[str]=0      )	->			Optional[Any]:


         '''simple docstring'''





         if curr_node:
                preorder.append((curr_node.val, level)      )
                self.__traversal(curr_node.left       ,       lowercase_       ,       level + 1      )
                self.__traversal(curr_node.right       ,       lowercase_       ,       level + 1      )
         else:
                preorder.append(('''#''', level)      )




  def __str__( self					: str      )	->			Dict:


         '''simple docstring'''





         if self.is_empty():
                return ""
         __magic_name__			:		Union[str, Any]        =     self.pre_order()

         return "\n".join(('''-''' * level + str(lowercase_      )) for value, level in preorder_heap      )


# Unit Tests
if __name__ == "__main__":
 import doctest

 doctest.testmod()





"""simple docstring"""


def 					_lowerCAmelCase (  lowercase_		):
						return " ".join(sorted(set(sentence.split()		)		)		)


if __name__ == "__main__":
				import doctest

				doctest.testmod()





