
"""simple docstring"""







class 				UpperCAmelCase_	:
 """simple docstring"""




 def __init__( self       : List[Any]     ,   a_       : int )->					Union[str, Any]:



      """simple docstring"""

      UpperCAmelCase_						:    Union[str, Any]    					=  val
      # Number of nodes in left subtree
      UpperCAmelCase_						:    Dict    					=  0
      UpperCAmelCase_						:    str    					=  None
      UpperCAmelCase_						:    int    					=  None
      UpperCAmelCase_						:    Dict    					=  None






 def   a      ( self       : Optional[int]     ,   a_       : List[Any] )->					str:



      """simple docstring"""

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

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



class 				UpperCAmelCase_	:
 """simple docstring"""




 def __init__( self       : List[str]     ,   a_       : Optional[int]=None     ,   a_       : str=None     ,   a_       : List[Any]=0 )->					Union[str, Any]:



      """simple docstring"""

      UpperCAmelCase_						:    Any    					=  heap_size
      UpperCAmelCase_						:    int    					=  bottom_root
      UpperCAmelCase_						:    Any    					=  min_node




 def   a      ( self       : int     ,   a_       : int )->					str:



      """simple docstring"""

      if other.size == 0:
           return None
      if self.size == 0:
           UpperCAmelCase_						:    List[str]    					=  other.size
           UpperCAmelCase_						:    Tuple    					=  other.bottom_root
           UpperCAmelCase_						:    Tuple    					=  other.min_node
           return None
      # Update size
      UpperCAmelCase_						:    Optional[Any]    					=  self.size + other.size

      # Update min.node
      if self.min_node.val > other.min_node.val:
           UpperCAmelCase_						:    Optional[Any]    					=  other.min_node
      # Merge

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

                # Merging trees
                UpperCAmelCase_						:    str    					=  i.merge_trees(i.parent )

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

      # Update other
      UpperCAmelCase_						:    Optional[int]    					=  self.size
      UpperCAmelCase_						:    Optional[Any]    					=  self.bottom_root
      UpperCAmelCase_						:    str    					=  self.min_node

      # Return the merged heap
      return self




 def   a      ( self       : Any     ,   a_       : Dict )->					List[Any]:



      """simple docstring"""

      if self.size == 0:
           UpperCAmelCase_						:    Tuple    					=  Node(a_ )
           UpperCAmelCase_						:    Optional[int]    					=  1
           UpperCAmelCase_						:    Optional[int]    					=  self.bottom_root
      else:
           # Create new node
           UpperCAmelCase_						:    List[Any]    					=  Node(a_ )

           # Update size
           self.size += 1

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

                # Merge
                UpperCAmelCase_						:    int    					=  self.bottom_root.merge_trees(self.bottom_root.parent )

                # Update Links
                UpperCAmelCase_						:    Dict    					=  next_node
                UpperCAmelCase_						:    Optional[Any]    					=  None
                if next_node:
                     UpperCAmelCase_						:    Union[str, Any]    					=  self.bottom_root




 def   a      ( self       : Dict )->					Tuple:



      """simple docstring"""

      return self.min_node.val




 def   a      ( self       : int )->					Dict:



      """simple docstring"""

      return self.size == 0




 def   a      ( self       : List[Any] )->					Union[str, Any]:



      """simple docstring"""

      UpperCAmelCase_						:    Tuple    					=  self.min_node.val

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

           # Update bottom root
           UpperCAmelCase_						:    Any    					=  None

           # Update min_node
           UpperCAmelCase_						:    List[str]    					=  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
           UpperCAmelCase_						:    Union[str, Any]    					=  self.bottom_root.parent
           UpperCAmelCase_						:    Union[str, Any]    					=  None

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

      # Size, min_node and bottom_root
      while bottom_of_new.left:
           UpperCAmelCase_						:    Any    					=  size_of_new * 2 + 1
           UpperCAmelCase_						:    Tuple    					=  bottom_of_new.left
           if bottom_of_new.val < min_of_new.val:
                UpperCAmelCase_						:    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):
           UpperCAmelCase_						:    Union[str, Any]    					=  size_of_new
           UpperCAmelCase_						:    Union[str, Any]    					=  bottom_of_new
           UpperCAmelCase_						:    Tuple    					=  min_of_new
           # print("Single root, multiple nodes case")
           return min_value
      # Remaining cases
      # Construct heap of right subtree
      UpperCAmelCase_						:    Optional[Any]    					=  BinomialHeap(
          bottom_root=a_     ,   min_node=a_     ,   heap_size=a_ )

      # Update size
      UpperCAmelCase_						:    Union[str, Any]    					=  self.size - 1 - size_of_new

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

      # Initialize new bottom_root and min_node
      UpperCAmelCase_						:    Optional[int]    					=  previous_node or next_node
      UpperCAmelCase_						:    Tuple    					=  next_node

      # Update links of previous_node and search below for new min_node and
      # bottom_root
      if previous_node:
           UpperCAmelCase_						:    Dict    					=  next_node

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

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

      return min_value




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



      """simple docstring"""

      UpperCAmelCase_						:    List[Any]    					=  self.bottom_root
      while top_root.parent:
           UpperCAmelCase_						:    Tuple    					=  top_root.parent
      # preorder
      UpperCAmelCase_						:    str    					=  []
      self.__traversal(a_     ,   a_ )
      return heap_pre_order




 def   a      ( self       : Dict     ,   a_       : Tuple     ,   a_       : Dict     ,   a_       : List[str]=0 )->					Union[str, Any]:



      """simple docstring"""

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






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



      """simple docstring"""

      if self.is_empty():
           return ""
      UpperCAmelCase_						:    str    					=  self.pre_order()

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


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

       doctest.testmod()



"""simple docstring"""







from __future__ import annotations



class 		UpperCAmelCase_	:
 """simple docstring"""




 def __init__( self       : Any     ,   a_       : int = 0 )->					None:



      """simple docstring"""

      UpperCAmelCase_						:    Tuple    					=  value
      UpperCAmelCase_						:    Node | None    					=  None
      UpperCAmelCase_						:    Node | None    					=  None




def 			A_			(					lowercase						, lowercase      )     ->							Node | None:


     """simple docstring"""







     if treea is None:
          return treea
     if treea is None:
          return treea

     UpperCAmelCase_						:    int    					=  treea.value + treea.value
     UpperCAmelCase_						:    List[str]    					=  merge_two_binary_trees(treea.left						, treea.left      )
     UpperCAmelCase_						:    List[str]    					=  merge_two_binary_trees(treea.right						, treea.right      )
     return treea




def 			A_			(					lowercase      )     ->							None:


     """simple docstring"""







     if root:
          print(root.value      )
          print_preorder(root.left      )
          print_preorder(root.right      )


if __name__ == "__main__":
       lowercase_						    =					Node(1)
       lowercase_						    =					Node(2)
       lowercase_						    =					Node(3)
       lowercase_						    =					Node(4)

       lowercase_						    =					Node(2)
       lowercase_						    =					Node(4)
       lowercase_						    =					Node(6)
       lowercase_						    =					Node(9)
       lowercase_						    =					Node(5)

       print("Tree1 is: ")
       print_preorder(treea)
       print("Tree2 is: ")
       print_preorder(treea)
       lowercase_						    =					merge_two_binary_trees(treea, treea)
       print("Merged Tree is: ")
       print_preorder(merged_tree)

