let be an ordered dictionary with entries show how to modify the avl tree to implement the following method for in time (logn)countallinrange( , )compute and return the number of entries in with key such that < < - show that the nodes that become unbalanced in an avl tree after operation insertatexternal is performedwithin the execution of an insert operationmay be nonconsecutive on the path from the newly inserted node to the root - show that at most one node in an avl tree becomes unbalanced after operation removeexternal is performed within the execution of remove dictionary operation - show that at most one trinode restructuring operation is needed to restore balance after any insertion in an avl tree - let and be ( , trees storing and entriesrespectivelysuch that all the entries in have keys less than the keys of all the entries in describe an (logn logmtime method for joining tand into single tree that stores all the entries in and - repeat the previous problem for red-black trees and - justify proposition - the boolean indicator used to mark nodes in red-black tree as being "redor "blackis not strictly needed when we have distinct keys describe scheme for implementing red-black tree without adding any extra space to standard binary search tree nodes how does your scheme affect the search and update times