I'm looking to improve my intuition and visualization of what large countable trees look like and I've ran into the issue that I have no understanding of what a tree of height, say, $\omega^2$ looks like. I'll include pictures of the kind of thing I'm looking for, and where my understanding breaks down.
An $\omega$ tree has a subtree that's isomorphic to something that looks like this,
An $\omega+1$ subtree has a subtree isomorphic to something that looks like this,
Repeating this process (of adding a node at the top) and taking unions means I've got some handle on what a tree of height $\omega . 2$ looks like, it's a union of things like this.
I can understand how to keep going with this process and my imagination (maybe) stretches far enough as $\omega.3$, but I really can't get any further with it.
My question is, is there any nice ways of visualizing a tree of height $\omega ^2$ as a Hasse diagram like this? Or is trying to use your intuition on a problem like this kind of futile?