Isolation Forest Score Function Theory

I am currently reading this paper on isolation forests. In the section about the score function, they mention the following. For context, $h(x)$ is definded as the path length of a data point traversing an iTree, and $n$ is the sample size used to grow the iTree.

The difficulty in deriving such a score from $h(x)$
is that while the maximum possible height of iTree grows
in the order of $n$, the average height grows in the order of
$log(n)$. Normalization of $h(x)$ by any of the above terms
is either not bounded or cannot be directly compared.

So herein lies my first question. What do they mean that the normalization of $h(x)$ by any of the above terms is either not bounded or cannot be directly compared? The final score function in this paper is given by

$$s(x,n)=2^{-frac{E(h(x))}{c(n)}}$$

where

$$c(n)=2H_{n-1}-2(n-1)/n$$

is the average path length of an unsuccessful search from BST Theory. Note how they are taking the expectation of the path. So in that case we are averaging all the values anyway, so why can we not use the growth of the average height of the tree? Additionally they don’t even mention the analytical form of average height here, which from what I gather is $2sqrt{pi n}$ reference, though I have not read this reference thoroughly and may be mistaken.

Am I missing something here?