the implementation of skew heaps is left as (trivialexercise note that because right path could be longa recursive implementation could fail because of lack of stack spaceeven though performance would otherwise be acceptable skew heaps have the advantage that no extra space is required to maintain path lengths and no tests are required to determine when to swap children it is an open problem to determine precisely the expected right path length of both leftist and skew heaps (the latter is undoubtedly more difficultsuch comparison would make it easier to determine whether the slight loss of balance information is compensated by the lack of testing binomial queues although both leftist and skew heaps support merginginsertionand deletemin all effectively in (log ntime per operationthere is room for improvement because we know that binary heaps support insertion in constant average time per operation binomial queues support all three operations in (log nworst-case time per operationbut insertions take constant time on average binomial queue structure binomial queues differ from all the priority queue implementations that we have seen in that binomial queue is not heap-ordered tree but rather collection of heap-ordered treesknown as forest each of the heap-ordered trees is of constrained form known as binomial tree (the reason for the name will be obvious laterthere is at most one binomial tree of every height binomial tree of height is one-node treea binomial treebk of height is formed by attaching binomial treebk- to the root of another binomial treebk- figure shows binomial trees and from the diagram we see that binomial treebk consists of root with children bk- binomial trees of height have exactly nodesand the number of nodes at depth is the binomial coefficient dk if we impose heap order on the binomial trees and allow at most one binomial tree of any heightwe can represent priority queue of any size by collection of binomial trees for instancea priority queue of size could be represented by the forest we might write this representation as which not only represents in binary but also represents the fact that and are present in the representation and is not as an examplea priority queue of six elements could be represented as in figure binomial queue operations the minimum element can then be found by scanning the roots of all the trees since there are at most log different treesthe minimum can be found in (log ntime alternativelywe can maintain knowledge of the minimum and perform the operation in ( time if we remember to update the minimum when it changes during other operations merging two binomial queues is conceptually easy operationwhich we will describe by example consider the two binomial queuesh and with six and seven elementsrespectivelypictured in figure