Wold-type decompositions for norm-increasing $m$-concave weighted shifts on directed trees

In this article, we show that every norm-increasing $3$-concave weighted shift on a directed tree admits Wold-type decomposition. We then classify the $4$-isometries within a class of weighted shifts on a rootless directed tree introduced by S. Chavan and S. Trivedi, and use this classification to construct analytic norm-increasing strict $4$-isometries without the wandering subspace property. This answers a question of S. Shimorin in the negative for every $m\in {\mathbb{Z}}_{⩾4}$.