Tree approaches (binomial or trinomial trees) are very popularly used in finance industry to price financial derivatives. Such popularity stems from their simplicity and clear financial interpretation of the methodology. On the other hand, PDE (partial differential equation) approaches, with which standard numerical procedures such as the finite difference method (FDM), are characterized with the wealth of existing theory, algorithms and numerical software that can be applied to solve the problem. For a simple geometric Brownian motion model, the connection between these two approaches is studied, but it is lower-order equivalence. Moreover such a connection for a regime-switching model is not so clear at all. This paper presents the high-order equivalence between the two for regime-switching models. Moreover the convergence rates of trinomial trees for pricing options with state-dependent switching rates are first proved using the theory of the FDMs.