Subnormal weighted shifts on directed trees and composition operators in $L^2$ spaces with non-densely defined powers

It is shown that for every positive integer $n$ there exists a subnormal weighted shift on a directed tree (with or without root) whose $n$th power is densely defined while its $(n+1)$th power is not. As a consequence, for every positive integer $n$ there exists a non-symmetric subnormal composition operator $C$ in an $L^2$ space over a $\sigma$-finite measure space such that $C^n$ is densely defined and $C^{n+1}$ is not.