Recognizing Union-Find trees is NP-complete

Disjoint-Set forests, consisting of Union-Find trees, are data structures having a widespread practical application due to their efficiency. Despite them being fundamental, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression). In this paper we provide such a characterization by means of a simple operation and we show that the problem of deciding whether a given tree is a Union-Find tree is NP-complete.