contains exactly nodesand binomial heap can only contain zero or one binomial tree of each orderso the total number of nodes in binomial heap must be nx bk bk [ = where bk specifies the number of trees of order thus there is one-to-one mapping between the binomial heap structure and the standard binary representation of the number nand since the binary representation is clearly uniqueso is the binomial heap structure the maximum number of trees in heap with nodes therefore equals the number of digits when is written in binary without leading zerosi log the heap can be stored efficiently as linked list of root nodes ordered by increasing tree order the most important operation for binomial heaps is mergebecause that can be used as sub-process for most other operations underlying that is the merge of two binomial trees of order into binomial tree of order by definitionthat is achieved by adding one of those trees as the left most sub-tree of the root of the otherand preservation of the priority ordering simply requires that it is the tree with the highest priority root that provides the root of the combined tree this clearly has ( time complexity then merging two whole binomial heaps is achieved by merging the constituent trees whenever there are two of the same orderin sequential manner analogous to the addition of two binary numbers in this casethe ( insert complexity will be multiplied by the number of treeswhich is (log )so the overall time complexity of merge is (log nthis is better than the (ncomplexity of merging binary heaps that can be achieved by concatenating the heap arrays and using the (nheapify algorithm insertion of new element into an existing binomial heap can easily be done by treating the new element as binomial heap consisting of single node ( an order zero tree)and merging that using the standard merge algorithm the average time complexity of that insert is given by computing the average number of ( tree combinations required the probability of needing the order zero combination is the probability of needing second combination is and the third is and so onwhich sum to one so insertion has ( overall time complexity that is better than the (log ncomplexity of insertion into standard binary heap creating whole new binomial heap from scratch can be achieved by using the ( insert process for each of the itemsgiving an overall time complexity of (nin this casethere is no better processso heapify here has the same time complexity as the heapify algorithm for binary heaps another important heap operation in practice is that of updating the heap after increasing node priority for standard binary heapsthat simply requires application of the usual bubble-up process with (log ncomplexity clearlya similar process can be used in binomial heapsand that will also be of (log ncomplexity the highest priority node in binomial heap will clearly be the highest priority root nodeand pointer to that can be maintained by each heap update operation without increasing the complexity of the operation serving the highest priority item requires deleting the highest priority node from the order tree it appears inand that will break it up into another binomial heap consisting of trees of all orders from to howeverthose trees can easily be merged back into the original heap using the standard merge algorithmwith the