Radical Dynamical Monogenicity

Let $a$ be an integer and $p$ a prime so that $f(x)=x^p-a$ is irreducible. Write $f^n(x)$ to indicate the $n$-fold composition of $f(x)$ with itself. We study the monogenicity of number fields defined by roots of $f^n(x)$ and give necessary and sufficient conditions for a root of $f^n(x)$ to yield a power integral basis for each $n\geq 1$.