
'''simple docstring'''
class 				UpperCAmelCase_			:





 def __init__(      self	,    lowercase_):
     snake_case_	:      Optional[int]            =  val
     # Number of nodes in left subtree
     snake_case_	:      List[str]            =  0
     snake_case_	:      Tuple            =  None
     snake_case_	:      int            =  None
     snake_case_	:      Dict            =  None





 def     snake_case__						(      self	,    lowercase_):
     assert self.left_tree_size == other.left_tree_size, "Unequal Sizes of Blocks"

     if self.val < other.val:
         snake_case_	:      Any            =  self.right
         snake_case_	:      Any            =  None
         if self.right:
             snake_case_	:      Tuple            =  other
         snake_case_	:      Tuple            =  other
         snake_case_	:      int            =  self.left_tree_size * 2 + 1
         return self
     else:
         snake_case_	:      str            =  other.right
         snake_case_	:      str            =  None
         if other.right:
             snake_case_	:      List[str]            =  self
         snake_case_	:      List[str]            =  self
         snake_case_	:      int            =  other.left_tree_size * 2 + 1
         return other






class 				UpperCAmelCase_			:





 def __init__(      self	,    lowercase_=None	,    lowercase_=None	,    lowercase_=0):
     snake_case_	:      List[str]            =  heap_size
     snake_case_	:      Optional[Any]            =  bottom_root
     snake_case_	:      Optional[Any]            =  min_node





 def     snake_case__						(      self	,    lowercase_):



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

     # Update min.node
     if self.min_node.val > other.min_node.val:
         snake_case_	:      List[str]            =  other.min_node
     # Merge

     # Order roots by left_subtree_size
     snake_case_	:      str            =  []
     snake_case_     ,	snake_case_	:      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))
             snake_case_	:      int            =  i.parent
         else:
             combined_roots_list.append((j, False))
             snake_case_	:      Optional[Any]            =  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]:
             snake_case_	:      str            =  combined_roots_list[i + 1][0]
             snake_case_	:      str            =  combined_roots_list[i][0]
        # Consecutively merge roots with same left_tree_size
     snake_case_	:      List[str]            =  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
             snake_case_	:      Any            =  i.left
             snake_case_	:      Optional[Any]            =  i.parent.parent

             # Merging trees
             snake_case_	:      List[Any]            =  i.merge_trees(i.parent)

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

     # Update other
     snake_case_	:      Optional[int]            =  self.size
     snake_case_	:      int            =  self.bottom_root
     snake_case_	:      Dict            =  self.min_node

     # Return the merged heap
     return self





 def     snake_case__						(      self	,    lowercase_):



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

         # Update size
         self.size += 1

         # update min_node
         if val < self.min_node.val:
             snake_case_	:      Union[str, Any]            =  new_node
         # Put new_node as a bottom_root in heap
         snake_case_	:      str            =  new_node
         snake_case_	:      int            =  self.bottom_root
         snake_case_	:      Dict            =  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
             snake_case_	:      Union[str, Any]            =  self.bottom_root.parent.parent

             # Merge
             snake_case_	:      Any            =  self.bottom_root.merge_trees(self.bottom_root.parent)

             # Update Links
             snake_case_	:      Dict            =  next_node
             snake_case_	:      Optional[int]            =  None
             if next_node:
                 snake_case_	:      Optional[int]            =  self.bottom_root





 def     snake_case__						(      self):
     return self.min_node.val





 def     snake_case__						(      self):
     return self.size == 0





 def     snake_case__						(      self):
     snake_case_	:      Any            =  self.min_node.val

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

         # Update bottom root
         snake_case_	:      int            =  None

         # Update min_node
         snake_case_	:      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
         snake_case_	:      int            =  self.bottom_root.parent
         snake_case_	:      Optional[Any]            =  None

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

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

     # Update size
     snake_case_	:      Optional[Any]            =  self.size - 1 - size_of_new

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

     # Initialize new bottom_root and min_node
     snake_case_	:      Optional[Any]            =  previous_node or next_node
     snake_case_	:      int            =  next_node

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

         # Update bottom_root and search for min_node below
         snake_case_	:      str            =  previous_node
         snake_case_	:      Optional[Any]            =  previous_node
         while self.bottom_root.left:
             snake_case_	:      Any            =  self.bottom_root.left
             if self.bottom_root.val < self.min_node.val:
                 snake_case_	:      List[Any]            =  self.bottom_root
     if next_node:
         snake_case_	:      Tuple            =  previous_node

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

     return min_value





 def     snake_case__						(      self):
     snake_case_	:      Dict            =  self.bottom_root
     while top_root.parent:
         snake_case_	:      List[Any]            =  top_root.parent
     # preorder
     snake_case_	:      int            =  []
     self.__traversal(lowerCamelCase__	,    lowerCamelCase__)
     return heap_pre_order





 def     snake_case__						(      self	,    lowercase_	,    lowercase_	,    lowercase_=0):



     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):



     if self.is_empty():
         return ""
     snake_case_	:      List[str]            =  self.pre_order()

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


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

  doctest.testmod()






'''simple docstring'''
import math

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      Tuple            =  math.log(2,						1_0    ) * number
    snake_case_	:      Dict            =  round((log_number - int(__SCREAMING_SNAKE_CASE    )),						1_5    )

    return difference

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE = 6_7_8_9_1_0    ):
    """simple docstring"""






    snake_case_	:      Any            =  9_0
    snake_case_	:      Optional[int]            =  0

    snake_case_	:      Union[str, Any]            =  math.log(1.23,						1_0    )
    snake_case_	:      Dict            =  math.log(1.24,						1_0    )
    snake_case_	:      List[Any]            =  0

    while position < number:
        snake_case_	:      Union[str, Any]            =  log_difference(__SCREAMING_SNAKE_CASE    )

        if difference >= upper_limit:
            power_iterator += 9_3

        elif difference < lower_limit:
            power_iterator += 1_9_6

        else:
            snake_case_	:      Optional[Any]            =  power_iterator
            power_iterator += 1_9_6
            position += 1

    return previous_power


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  print(f"""{solution() = }""")




