we can represent path from the root to given node of binary tree by means of binary stringwhere means "go to the left childand means "go to the right child for examplethe path from the root to the node storing ( ,win the heap of figure is represented by " design an (logn)-time algorithm for finding the last node of complete binary tree with nodesbased on the above representation show how this algorithm can be used in the implementation of complete binary tree by means of linked structure that does not keep reference to the last node - given heap and key kgive an algorithm to compute all the entries in with key less than or equal to for examplegiven the heap of figure and query the algorithm should report the entries with keys and (but not necessarily in this orderyour algorithm should run in time proportional to the number of entries returned - provide justification of the time bounds in table - tamarindo airlines wants to give first-class upgrade coupon to their top log frequent flyersbased on the number of miles accumulatedwhere is the total number of the airlinesfrequent flyers the algorithm they currently usewhich runs in (nlogntimesorts the flyers by the number of miles flown and then scans the sorted list to pick the top logn flyers describe an algorithm that identifies the top logn flyers in (ntime - develop an algorithm that computes the kth smallest element of set of distinct integers in ( klogntime