  A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called
\emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and
$\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of
integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Define ${\mathcal
F}(x):=x^m+Ax^{m-1}+B$. In this article, we determine sets of conditions on
$m$, $A$, and $B$, such that the power-compositional trinomial ${\mathcal
F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$.
Furthermore, we prove the actual existence of infinite families of such
trinomials ${\mathcal F}(x)$.
