We propose new antichain-based algorithms for checking universality and inclusion of nondeterministic tree automata (NTA). We have implemented these algorithms in a prototype tool and our experiments show that they provide a significant improvement over the traditional determinisation-based approaches. We use our antichain-based inclusion checking algorithm to build an abstract regular tree model checking framework based entirely on NTA. We show the significantly improved efficiency of this framework through a series of experiments with verifying various programs over dynamic linked tree-shaped data structures.