  For certain pairs of unimodal maps on the interval with periodic critical
orbits, it is known that one can combine them to create another map whose
entropy is close to one while the poles of Artin-Mazur $\zeta$ function outside
the unit circles can be made close to the other. We provided a formulation and
proof of this result in symbolic dynamics setting, which allow us to generalize
this fact to certain families of maps on finite trees.
