PA is the process algebra allowing non-determinism, sequential and parallel compositions, and recursion. We suggest a view of PA-processes as {\em tree languages}. Our main result is that the set of iterated predecessors of a regular set of PA-processes is a regular tree language, and similarly for iterated successors. Furthermore, the corresponding tree-automata can be built effectively in polynomial-time. This has many immediate applications to verification problems for PA-processes, among which a simple and general model-checking algorithm.