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?

heightisn't really the right term for your usage here, since in set theory when we speak of a tree of height $\alpha$, we usually mean that it can be stratified into $\alpha$ many levels, with the predecessors of a node on lower levels. For example, a Suslin tree is a kind of tree of height $\omega_1$. What you are concerned with would usually be called therankof a well-founded tree. $\endgroup$ – Joel David Hamkins Aug 20 '13 at 16:00heightin my sense (the usual sense), then the trees grow upward, and the root is minimal. Any node on level $\omega$ or higher has a linear $\omega$-chain below it. So this is a very different meaning than the OP's. $\endgroup$ – Joel David Hamkins Aug 21 '13 at 19:01