  In this paper, we study hyperbolic rational maps with finitely connected
Fatou sets. We construct models of post-critically finite hyperbolic tree
mapping schemes for such maps, generalizing post-critically finite rational
maps in the case of connected Julia set. We show they are general limits of
rational maps as we quasiconformally stretch the dynamics. Conversely, we use
quasiconformal surgery to show any post-critically finite hyperbolic tree
mapping scheme arises as such a limit.
  We construct abundant examples thanks to the flexibilities of the models, and
use them to construct a sequence of rational maps of a fixed degree with
infinitely many non-monomial rescaling limits.
