- describe how to perform an operation remove range(startstopthat removes all the items whose keys fall within range(startstopin sorted map that is implemented with binary search tree and show that this method runs in time ( )where is the number of items removed and is the height of - repeat the previous problem using an avl treeachieving running time of ( log nwhy doesn' the solution to the previous problem trivially result in an ( log nalgorithm for avl treesc- suppose we wish to support new method count range(startstopthat determines how many keys of sorted map fall in the specified range we could clearly implement this in ( htime by adapting our approach to find range describe how to modify the search tree structure to support (hworst-case time for count range - if the approach described in the previous problem were implemented as part of the treemap classwhat additional modifications (if anywould be necessary to subclass such as avltreemap in order to maintain support for the new methodc- draw schematic of an avl tree such that single remove operation could require (log ntrinode restructurings (or rotationsfrom leaf to the root in order to restore the height-balance property - in our avl implementationeach node stores the height of its subtreewhich is an arbitrarily large integer the space usage for an avl tree can be reduced by instead storing the balance factor of nodewhich is defined as the height of its left subtree minus the height of its right subtree thusthe balance factor of node is always equal to - or except during an insertion or removalwhen it may become temporarily equal to - or + reimplement the avltreemap class storing balance factors rather than subtree heights - if we maintain reference to the position of the leftmost node of binary search treethen operation find min can be performed in ( time describe how the implementation of the other map methods need to be modified to maintain reference to the leftmost position - if the approach described in the previous problem were implemented as part of the treemap classwhat additional modifications (if anywould be necessary to subclass such as avltreemap in order to accurately maintain the reference to the leftmost positionc- describe modification to the binary search tree implementation having worst-case ( )-time performance for methods after(pand before(pwithout adversely affecting the asymptotics of any other methods