figure input for exercise prove that binomial tree of height has dk nodes at depth merge the two binomial queues in figure show that inserts into an initially empty binomial queue take (ntime in the worst case give an algorithm to build binomial queue of elementsusing at most comparisons between elements propose an algorithm to insert nodes into binomial queue of elements in ( log nworst-case time prove your bound write an efficient routine to perform insert using binomial queues do not call merge  for the binomial queue modify the merge routine to terminate merging if there are no trees left in and the carry tree is nullptr modify the merge so that the smaller tree is always merged into the larger suppose we extend binomial queues to allow at most two trees of the same height per structure can we obtain ( worst-case time for insertion while retaining (log nfor the other operations suppose you have number of boxeseach of which can hold total weight and items in which weigh wn respectively the object is to pack all the items without placing more weight in any box than its capacity and using as few boxes as possible for instanceif and the items have weights then we can solve the problem with two boxes in generalthis problem is very hardand no efficient solution is known write programs to implement efficiently the following approximation strategiesa place the weight in the first box for which it fits (creating new box if there is no box with enough room(this strategy and all that follow would give three boxeswhich is suboptimal place the weight in the box with the most room for it place the weight in the most filled box that can accept it without overflowing  are any of these strategies enhanced by presorting the items by weight