  Disjoint-Set forests, consisting of Union-Find trees, are data structures
having a widespread practical application due to their efficiency. Despite them
being well-known, no exact structural characterization of these trees is known
(such a characterization exists for Union trees which are constructed without
using path compression) for the case assuming union-by-rank strategy for
merging. In this paper we provide such a characterization by means of a simple
push operation and show that the decision problem whether a given tree (along
with the rank info of its nodes) is a Union-Find tree is NP-complete,
complementing our earlier similar result for the union-by-size strategy.
