The concept of power for monotone invariant clustering procedures is developed via the possible partitions of objects at each iteration level in the obtained hierarchy. At a given level, the probability of rejecting the randomness hypothesis is obtained empirically for the possible types of partitions of the n objects employed. The results indicate that the power of a particular hierarchical clustering procedure is a function of the type of partition. The additional problem of estimating a “true” partition at a certain level of a hierarchy is discussed briefly.