Prove that 9 is the minimum number of calls to make-set, union-set, find-set such that a disjoint set union using weight (number of nodes) by path compression and disjoint set union using rank (upper bound on height) such that the trees produced by this sequence of calls is different.
So far I've found a tree example for which this holds. But now I must prove that this is the minimum. My first idea is contradiction but I'm not sure how to use it well, or maybe some other argument works better. Any help would be appreciated!
The proof idea I have takes a tree from height 0 and then keeps growing until I realize it needs to have 9 operations. But I don't think this is enough.
