suppose two binary treest and hold entries satisfying the heap-order property describe method for combining and into tree whose internal nodes hold the union of the entries in andt and also satisfy the heap-order property your algorithm should run in time ( where and are the respective heights of and - give an alternative analysis of bottom-up heap construction by showing the following summation is ( )for any positive integer hc- give an alternate description of the in-place heap-sort algorithm that uses standard comparator instead of reverse one - describe efficient algorithms for performing operations remove(eand replacekey( ,kon an adaptable priority queue realized by means of an unsorted list with location-aware entries - describe efficient algorithms for performing operations remove(eand replacekey( ,kon an adaptable priority queue realized by means of heap with location-aware entries - let be set of points in the plane with distinct integer xand ycoordinates let be complete binary tree storing the points from at its external nodessuch that the points are ordered left-to-right by in creasing -coordinates for each node in tlet (vdenote the subset of consisting of points stored in the subtree rooted at for the root of tdefine top(rto be the point in (rwith maximum -coordinate for every other node vdefine top(rto be the point in with highest -coordinate in (vthat is not also the highest ycoordinate in ( )where is the parent of in (if such point existssuch labeling turns into priority search tree describe linear-time algorithm for turning into priority search tree projects