Does $f(n,z)$ have $2^n$ distinct fixpoints $z$ for all $n$?

Question

Let $f(z)$ be a given degree $2$ polynomial. Let $n$ be a positive integer. Let $f(1,z) = f(z)$ and $f(n,z)= f(1,f(n-1,z))$.

How to decide if $f(n,z)$ has $2^n$ distinct fixpoints $z$ for all $n$ ?