User blog comment:Hyp cos/tree function and TREE(3)/@comment-5529393-20140625041914/@comment-11227630-20140625051100

In part 1 and 2 the tables showed my "natural ordering". Higher-ordered trees aren't homeomorphical embeddable in lower-ordered trees. For example, the () has level $$\varepsilon_0$$, and it's not homeomorphical embeddable in any binary trees, which all have level smaller than $$\varepsilon_0$$.

Sometimes a lower-ordered tree isn't homeomorphical embeddable in some higher-ordered trees, and that will be some improvement chance. The tree become ([(())][][]) (a 3-children tree above [(())]) in step 7, and it's lower-ordered than ([(())][]) (a 4-children tree above [(())]), but they're not homeomorphical embeddable in each other, and again ([(())]) (here I just get 1 extra step). Then back to ([(())]T) with T<[] (i.e. T is made up of 's). When it reduces to ([(())]), it goes back to a 3-children tree above [(())] - that's ([(())][]()).

In step 6 the [((()))] reduces to (([(())])) rather than ([(())][]); also in step 25 the [()] reduces to (([])) rather than ([][]). Those are the results of some comparisons, but I really don't know why. Maybe there're some ways that make TREE(n) stronger than $$H_{\vartheta(\Omega^\omega\omega)}(n)$$, but I just don't know that.