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 $\bar{M}_K$. Consider a power series $f(x) \in O_K[[x]]$ without constant term. 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, $A$ is the set of Torsion points of $f$ of exactly $n$-th iteration while $B$ is the set preperiodic points of $f$.
My question:
$(1)$ when is there a bijective correspondence between the sets $A_n$ and $B_{n,m}$ on some condition of $m$?
$(2)$ Is there a bijective correspondence between the sets $A_n$ and $C_n$ ?