2. (20 points) A maxtree data structure on n = 2'\" elements is a complete binary tree data structure of depth k With 71 leaves, Where the leaves store integers and the interior nodes store the maximum value of any leaf in their sub-tree. We View the leaves as being indexed from 1 to n, from left to right. For each of the operations given below, give a 0(k) = 0(logn) time algorithm for performing the operation on a maxtree. You should give an explanation for your algorithms, but don't need a formal proof of correctness or pseudocode: (a) Given the following information: index i such that 1 g 'i S n, positive integer v and a pointer to the root 7" of a Max-tree of size 71, Describe an algorithm to add v to the current value of the i'th leaf and update the appropriate interior nodes so that it follows the rules of being a max-tree. (b) Given the following information: index i such that 1 S 2' S n, positive integer v and a pointer to the root 7' of a Maxtree of size n, Describe an algorithm that returns the maximum value of the leaves 1 through