priority queues (heapsa binary tree that has two extra levels (that isit is very slightly unbalancedb binary tree that has deepest node at depth log binary tree that has deepest node at depth log the worst-case binary tree rewrite the binaryheap insert routine by placing copy of the inserted item in position how many nodes are in the large heap in figure prove that for binary heapsbuildheap does at most - comparisons between elements show that heap of eight elements can be constructed in eight comparisons between heap elements  give an algorithm to build binary heap in (log nelement compar isons show the following regarding the maximum item in the heapa it must be at one of the leaves there are exactly  / leaves every leaf must be examined to find it  show that the expected depth of the kth smallest element in large complete heap (you may assume is bounded by log give an algorithm to find all nodes less than some valuexin binary heap your algorithm should run in ( )where is the number of nodes output does your algorithm extend to any of the other heap structures discussed in this give an algorithm that finds an arbitrary item in binary heap using at most roughly / comparisons  propose an algorithm to insert nodes into binary heap on elements in ( log log log ntime prove your time bound write program to take elements and do the followinga insert them into heap one by one build heap in linear time compare the running time of both algorithms for sortedreverse-orderedand random inputs each deletemin operation uses log comparisons in the worst case propose scheme so that the deletemin operation uses only log log log ( comparisons between elements this need not imply less data movement  extend your scheme in part (aso that only log log log log ( comparisons are performed  how far can you take this idea do the savings in comparisons compensate for the increased complexity of your algorithmif -heap is stored as an arrayfor an entry located in position iwhere are the parents and children