This seems like an excellent simplification to me.
The one weird thing that occurs is that people often talk about
going "up" the tree as towards the root, which is the opposite of a
real world tree. E.g., depth first search. So I'd suggest we say
Up: towards the root, decreasing rank
Down: towards the leaves, increasing rank
Higher node: lower rank
Lower node: higher rank
I think it's important to define higher and lower in terms of Rank
since Rank is what's actually used. Otherwise you get into a weird
situation comparing disjoint subtrees.
