Ahlfors regular spaces have regular subspaces of any dimension

We characterize $Q$-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each $0 < \alpha < Q$, an $\alpha$-regular subspace. As an application, we give an alternative simple proof of the existence of $\alpha$-regular subspaces of a $Q$-dimensional complete Ahlfors regular metric space $(X,\rho)$, which was proved in \cite{JJKRRS}.