Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest degree coefficient is unit, otherwise it is non-invertible. Consider a non-invertible power series $f(x) \in O_K[[x]]$ without constant term of weierstrass degree $\text{wideg}(f)=p^r$, $r$ is a natural number. Now consider the following two sets (for fixed $n$):
\begin{align*}
A_n&=\{x \in \bar{M}_K ~|~ f^n(x)=0 \ \text{and} \ f^{n-1}(x) \neq 0, \ n \geq 1 \}, \\
B_{n,m}&=\{x \in \bar{M}_K ~|~ f^n(x)=f^m(x), \ \text{where} \ n>m \geq 0 \},\\
C_n&=\{x \in \bar{M}_K ~|~ f^n(x)=f^{n-1}(x) \}.
\end{align*}
Here $f^n=f \circ f \circ \cdots \circ f$ ($n$ times) is the iteration of $f$.
In fact, $A_n$ is the set of Torsion points of $f$ of exactly $n$-th iteration while $B_{n,m}$ and $C_n$ contains preperiodic points of $f$.
Question:
When is there a bijective correspondence between the sets $A_n$ and $B_{n,m}$ on some condition of $m$? Is there a bijective correspondence between the sets $A_n$ and $C_n$ ?
