The single equality $A^{\ast n}{A}^n=(A^{\ast }A{)}^n$ does not imply the quasinormality of weighted shifts on rootless directed trees

It is proved that each bounded injective bilateral weighted shift $W$ satisfying the equality $W^{\ast n}{W}^n=(W^{\ast }W{)}^n$ for some integer $n\ge 2$ is quasinormal. For any integer $n\ge 2$, an example of a bounded non-quasinormal weighted shift $A$ on a rootless directed tree with one branching vertex which satisfies the equality $A^{\ast n}{A}^n=(A^{\ast }A{)}^n$ is constructed. It is also shown that such an example can be constructed in the class of composition operators in $L^2$-spaces over $\sigma$-finite measure spaces.