While the ideas used in disjoint-set forests have long been familiar, Robert Tarjan was the first to prove the upper bound (and a restricted version of the lower bound) in terms of the inverse Ackermann function, in 1975.[5] Until this time the best bound on the time per operation, proven by Hopcroft and Ullman,[6] was O(log* n), the iterated logarithm of n, another slowly growing function (but not quite as slow as the inverse Ackermann function).