Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a selfsimilar
set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that
under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In
the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such
trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected
self-similar sets which has been undergoing some extensive development recently.