  In this paper, we study quasi post-critically finite degenerations for
rational maps. We construct a limit for such degeneration as a geometrically
finite rational map on a finite tree of Riemann spheres. We prove the
boundedness for such degenerations of hyperbolic rational maps with Sierpinski
carpet Julia set and characterize the convergence for quasi-Blaschke products
$\mathcal{QB}_d$. These are natural analogs of Thurston's compactness theorem
for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian
groups. In the appendix, we apply such convergence results to show the
existence of certain polynomial matings.
