User blog:Hyp cos/tree function and TREE(3)

We know the weak tree function has growth rate \(\vartheta(\Omega^\omega)\) in FGH or HH. However, to find a good bound of TREE(3), it's not enough that we just know tree(n) is comparable to \(H_{\vartheta(\Omega^\omega)}(n)\). We should know how it works. Here I found some results about tree function. We need a definition here for the second question.
 * Sequences whose length comparable to \(H_{\vartheta(\Omega^\omega)}(n)\) (maybe not the winning sequences)
 * A better bound for growth rate of TREE(n)
 * A sequence of TREE(3) and a bound

tree(n)
Sequences started with (...) can have length comparable to \(H_{\vartheta(\Omega^\omega)}(n)\). Now vertices have at most n children. Then it'll reduce to "all vertices have at most n-1 children", then "all vertices have at most n-2 children". Finally become a binary tree (but not ordered). It's at \(\varepsilon_0\) level (in HH).

If we get () and at most n vertices, the next tree is ((...()...)) with n+1 vertices. At the end we get and at most 2n+1 vertices, at the same time \(H_{\omega}(n)=2n\), so we call () has level \(\omega\).

More comparisons: