hlogn justificationfrom the fact that is completewe know that the number of nodes of is at least - this lower bound is achieved when there is only one node on level in additionalso following from being completewe have that the number of nodes of is at most this upper bound is achieved when level has nodes since the number of nodes is equal to the number of entrieswe obtain < and < + thusby taking logarithms of both sides of these two inequalitieswe see that <log and log( < since is an integerthe two inequalities above imply that hlogn proposition has an important consequencefor it implies that if we can perform update operations on heap in time proportional to its heightthen those operations will run in logarithmic time let us therefore turn to the problem of how to efficiently perform various priority queue methods using heap complete binary trees and their representation let us discuss more about complete binary trees and how they are represented