Let $${f \in \mathbb{Z}[x]}$$ , $${\deg f =3}$$ . Assume that$f$does not have repeated roots. Assume as well that, for every prime$q$, $${f(x)\not\equiv 0}$$ mod$q$^{2}has at least one solution in $${(\mathbb{Z}/q^2 \mathbb{Z})^*}$$ . Then, under these two necessary conditions, there are infinitely many primes$p$such that$f$($p$) is square-free.

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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.