We consider the extension of term rewriting rules with context constraints restricting the application of rewriting to positions whose prefix (i.e. the sequence of symbols from the rewrite position up to the root) belongs to a given regular language. This approach, well studied in the case of string rewriting, is similar to node selection mechanisms in XML transformation languages, and also generalizes the context-sensitive rewriting. The systems defined this way are called prefix constrained TRS (pCTRS), and we study the decidability of reachability of regular tree model checking and the preservation of regularity for some subclasses. The two latter properties hold for linear and right-shallow standard TRS but not any-more when adding context constraints. We show that these properties can be restored by restricting derivations to bottom-up ones, and moreover that it implies that left-linear and right-ground pCTRS preserve regularity and have a decidable regular model checking problem.