figure combining the binomial trees into binomial queue theorem the amortized running times of merge and insert are both ( for lazy binomial queues the amortized running time of deletemin is (log nproof the potential function is the number of trees in the collection of binomial queues the initial potential is and the potential is always nonnegative thusover sequence of operationsthe total amortized time is an upper bound on the total actual time for the merge operationthe actual time is constantand the number of trees in the collection of binomial queues is unchangedsoby equation ( )the amortized time is ( for the insert operationthe actual time is constantand the number of trees can increase by at most so the amortized time is ( the deletemin operation is more complicated let be the rank of the tree that contains the minimum elementand let be the number of trees thusthe potential at the start of the deletemin operation is to perform deleteminthe children of the smallest node are split off into separate trees this creates treeswhich must be merged into standard binomial queue the actual time to perform this is + +log