We introduce a new concept of Markov-type set-valued functions on trees allowing the graphs to be $2$-dimensional. Additionally, we present Markov set-valued functions on compact metric spaces. We establish the conditions under which two inverse limits of inverse sequences of trees or compact metric spaces are homeomorphic.

This is joint work with Iztok Banič and Matevž Črepnjak, both University of Maribor.