User blog:FundamentalSeq/Ordinal Tree Notation

While attempting to come up with simpler ways to understand TON and thinking about tree recursion, I ended up with this system, which I call Ordinal-Tree Notation, or OTN, which I hope maintains a lot of the expressiveness of TON while being a bit simpler to understand.

The Notation
Almost any tree notation will be valid, but the one which makes comparison easiest is as follows:

A node with n children is expressed as [n](child 1)(child 2)..., where [n] is "0" if the node has 0 children, and the n'th letter of the alphabet otherwise.

Comparison is then the standard lexicographic ordering on strings, with "0" < "A" < "B" < "C" < ...

Standard Forms
This notation borrows a few concepts from TON, including the idea of systems.

A tree is standard in the n'th system if either it is 0 or every path from the root passes at most n increments (edges of the form a -> b with a < b) before encountering a smaller standard tree.

For instance, in the first system:
 * AA0 is standard, because the only path immediately meets A0, a smaller standard tree.
 * ABB000 is standard, because the two paths are ABB000 < BB000 -> B00 -> 0 and ABB000 < BB000 -> 0, both of which end in a smaller standard tree (0) and contain one increment.
 * ABB00ABB00B00 is not standard, because of the path ABB00ABB00B00 < ABB00B00 < BB00.

You can check standardness here; change line 35 to do systems others than first.

Analysis
0'th system is fairly straightforward: the n'th child from the right of a node roughly corresponds to phi(n-1), and thus the system has limit phi(w,0). I'm not very good at first system, but here's an initial analysis reaching e0: