Unitary equivalence of balanced weighted shifts on rooted directed trees

We completely characterize non-periodic balanced weighted shifts $S_{\text{lambdab}}$ on rooted directed trees under a very mild assumption that $S_{\text{lambdab}}^{\ast n}{S}_{\text{lambdab}}^n{\mathrm{\mid }}_{\ker S_{\text{lambdab}}^{\ast }}$ is invertible operator on $\ker S_{\text{lambdab}}^{\ast }$ for all $n\in \mathbb{N}$. This generalizes the previously established unitary equivalences for Bergman and Dirichlet type shifts associated with locally finite rooted directed trees. We also give a counter example to justify that the criteria obtained for non-periodic balanced weighted shifts is not necessary for eventually periodic balanced weighted shifts.