approachthen that may look like better choice of approach than the third approach howevermaketree will itself generally be an (nprocedure if the trees are array-basedrather than pointer-basedwhich they usually are for binary heap trees sofor array-based similarly-sized binary heapsthe third approach is usually best if the heap trees to be merged have very different sizes and nthe first approach will have overall time complexity (mlog )which could be more efficient than an (napproach if in practicea good general purpose merge algorithm would check the sizes of the two trees and use them to determine the best approach to apply binomial heaps binomial heap is similar to binary heap as described abovebut has the advantage of more efficient procedures for insertion and merging unlike binary heapwhich consists of single binary treea binomial heap is implemented as collection of binomial trees definition binomial tree is defined recursively as followsa binomial tree of order is single node binomial tree of order has root node with children that are roots of binomial trees of orders (in that orderthusa binomial tree of order has height kcontains nodesand is trivially constructed by attaching one order - binomial tree as the left-most child of another order - binomial tree binomial trees of order and take the formand it is clear from these what higher order trees will look like binomial heap is constructed as collection of binomial trees with particular structure and node ordering propertiesthere can only be zero or one binomial tree of each order each constituent binomial tree must satisfy the priority ordering propertyi each node must have priority less than or equal to its parent