  Let $f$ be a polynomial over a global field $K$. For each $\alpha$ in $K$ and
$N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_N(f,\alpha)$ the arboreal field
$K(f^{-N}(\alpha))$ and by $D_N(f,\alpha)$ its degree over $K$.
  It is conjectured that $D_N(f,\alpha)$ should grow as a double exponential
function of $N$, unless $f$ is post-critically finite (PCF), in which case
there are examples like $D_N(x^2,\alpha) \leq 4^{N}$. There is evidence
conditionally on Vojta's conjecture. However, before the present work, no
unconditional non-trivial lower bound was known for post-critically infinite
$f$. In the case $f$ is PCF, no non-trivial lower bound was known, not even
under Vojta's conjecture.
  In this paper we give two simple methods that turn the finiteness of the
critical orbit into an exploitable feature, also in the post-critically
infinite case. First, assuming GRH for number fields, we establish for all PCF
polynomials $f$ of degree at least $2$ and all $\alpha$ outside of the critical
orbits of $f$, the existence of a positive constant $c(f,\alpha)$ such that
$$D_N(f,\alpha) \geq \text{exp}(c(f,\alpha)N), $$ for all $N$, which is sharp
up to, possibly, improve the constant $c(f,\alpha)$.
  Second, we show unconditionally that if $f$ is post-critically infinite over
any number field $K$ and unicritical, then, for each $\alpha$ in $K$, there
exists a positive constant $c(f,\alpha)$ such that $$D_N(f,\alpha) \geq
\text{exp}(c(f,\alpha)N), $$ for all $N$. The main input here is to work modulo
a suitably chosen prime and use a construction available for PCF unicritical
polynomials with periodic critical orbit.
