  In \cite{LuLa13}, two of the authors initiated a study of Lipschitz
equivalence of self-similar sets through the augmented trees, a class of
hyperbolic graphs introduced by Kaimanovich \cite{Ka03} and developed by Lau
and Wang \cite{LaWa09}. In this paper, we continue such investigation. We
remove a major assumption in the main theorem in \cite{LuLa13} by using a new
notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of
any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then
apply this result to consider the Lipschitz equivalence of certain totally
disconnected self-similar sets as well as their unions.
