recursion in figure by considering only half the graphyou can see that copies are necessary for an array of items (steps and )and copies are necessary for items similar calculations provide the number of copies necessary for larger arrays table summarizes this information table number of operations when is power of log number of copies into workspace ( *log ( ( ( ( ( ( ( total copies comparisons max (minactuallythe items are not only copied into the workspacethey're also copied back into the original array this doubles the number of copiesas shown in the total copies column the final column of table shows comparisonswhich we'll return to in moment it' harder to calculate the number of copies and comparisons when is not multiple of but these numbers fall between those that are power of for itemsthere are total copiesand for items , total copies number of comparisons in the mergesort algorithmthe number of comparisons is always somewhat less than the number of copies how much lessassuming the number of items is power of for each individual merging operationthe maximum number of comparisons is always one less than the number of items being mergedand the minimum is half the number of items being merged you can see why this is true in figure which shows two possibilities when trying to merge two arrays of four items each in the first casethe items interleaveand seven comparisons must be made to merge them in the second caseall the items in one array are smaller than all the items in the otherso only four comparisons need be made there are many merges for each sortso we must add the comparisons for each one referring back to figure you can see that seven merge operations are required to sort eight items the number of items being merged and the resulting number of comparisons are shown in table