Rational maps often support invariant conformal structures on their Fatou sets and can be deformed by quasiconformal mappings. If we focus on the stretching deformation of invariant collection of annuli arising from attracting and super attracting basins, Siegel disks and Herman rings, the limit of the deformation usually degenerates to lower degree rational maps. We introduce an R-tree associated with such deformation together with a piecewise linear mapping on the tree, which describes the dynamics on the Fatou components, and discuss connections with quasiconformal surgery, etc.