I'm looking at the implementation of UnionFind with union by rank and path compression from here http://en.wikipedia.org/wiki/Disjoint-set_data_structure#Disjoint-set_forests (it's pretty much the same pseudo-code as in CLRS) and don't understand why path compression doesn't change rank. If we call find for an endpoint of the longest path from the root the rank should go down and if it doesn't the next union operation will choose an incorrect root.
"Rank" is one of those horribly overloaded terms in theoretical computer science. As Wikipedia notes, in the context of that disjoint set data structure with path compression, rank is not an intrinsic property of the current topology of the forest -- there's just no good way to keep the height of each node up to date. As defined by the sequence of unions, however, rank is useful in proving the running time bound involving the inverse Ackermann function.