figure converted to the leftist heap figure decreasekey( completed by merging and the same sizeso it violates the binomial queue property we will call this lazy binomial queue in order to maintain consistency this is fast operation that always takes constant (worst-casetime as beforean insertion is done by creating one-node binomial queue and merging the difference is that the merge is lazy the deletemin operation is much more painfulbecause it is where we finally convert the lazy binomial queue back into standard binomial queuebutas we will showit is still (log namortized time--but not (log nworst-case timeas before to perform deleteminwe find (and eventually returnthe minimum element as beforewe delete it from the queuemaking each of its children new trees we then merge all the trees into binomial queue by merging two equal-sized trees until it is no longer possible as an examplefigure shows lazy binomial queue in lazy binomial queuethere can be more than one tree of the same size to perform the deleteminwe remove the smallest elementas beforeand obtain the tree in figure we now have to merge all the trees and obtain standard binomial queue standard binomial queue has at most one tree of each rank in order to do this efficientlywe must