Wold-type decomposition for left-invertible weighted shifts on a rootless directed tree

Let $S_{\boldsymbol \lambda }$ be a bounded left-invertible weighted shift on a rootless directed tree $\mathcal T=(V, \mathcal E).$ We address the question of when $S_{\boldsymbol \lambda }$ has Wold-type decomposition. We relate this problem to the convergence of the series $$ \sum _{n = 1}^{\infty } \sum _{u \in G_{v, n}\setminus G_{v, n-1}} \bigg(\frac{{\boldsymbol \lambda }^{(n)}(u)}{{\boldsymbol \lambda }^{(n)}(v)}\bigg)^2,\quad v \in V, $$ involving the moments ${\boldsymbol \lambda }^{(n)}$ of $S^*_{\boldsymbol \lambda }$, where $G_{v, n}=\mathsf{Chi}^{\langle n\rangle } ({\mathsf{par}}^{\langle n\rangle }(v))$. Our main result characterizes all bounded left-invertible weighted shifts $S_{\boldsymbol \lambda }$ on $\mathcal T$ which have Wold-type decomposition.