TREE sequence

The TREE sequence is a very fast-growing function arising out of.

Suppose we have a sequence of k- T1, T2 ... with the following properties:


 * 1) Each tree Ti has at most i vertices.
 * 2) No tree is  into any tree following it in the sequence.

Explanation
The second property is confusing for non-experts. Since drawing trees in the natural form is impractical, we can represent tree as a set of parenthesis, where every represents a vertex, and all content which embedded to it means that this content is a children of that vertex. We can use to represent the 1-labeled vertices and [] to 2-labeled. For examples, [([])] means that we have 2-labeled vertex connected to two 1-labeled vertices, and one of them itself connected to other 2-labeled vertex.

Then, using this notation, what is the meaning of homeomorphic embedding? We can say that one tree is embedded to other one if we can find first entirely in the second, e.g. ([]) is embedded to [([])]. However, by this definition, say, ([][]) is also embedded to ([]([])). That's not correct with homeomorphic embedding. To make tree that homeomorphically embeddable to the other, we need that every part of the first tree will be able to find corresponding vertex in the second tree at the same "level of nesting". Here, level of nesting is a path from the root tree to the vertex. For example, in the tree ((([]))) the vertex [] has level of nesting 3, since we need to travel through three 's to get to [].

Known bounds
Returning to the function, states that sequence that has properties that assigned above cannot be infinite. Harvey Friedman expanded on this by asking the question: given k, what is the maximum length of such a sequence?

This maximal length is a function of k, dubbed TREE(k). The first two values are TREE(1) = 1 and TREE(2) = 3. The next value, TREE(3), is so large that it surpasses many large combinatorial constants like n(4); it is the subject of extensive research regarding its size. Since TREE(3) > n(4), a very weak lower bound for it is AA(187196)(1) using the Ackermann function. It has also been proven that TREE(3) > (5) for any "reasonable amount" X.

It can be shown that TREE(n) grows faster than \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy. The current limit of BEAF is believed to be around \(f_{\varphi(2,0,0)}(n)\), so TREE(3) probably surpasses many or all of Bowers' numbers, including meameamealokkapoowa oompa. It is certainly much smaller than Rayo's number, however &mdash; TREE(n) is still a computable function.

Chris Bird has shown that \(\text{TREE}(3) > \lbrace3, 6, 3 [1 [1 \neg 1,2] 2] 2\rbrace\), using his array notation.

Weak tree function
Weak tree function, tree(n) is defined to be longest sequence of 1-labelled trees such that:


 * 1) Every tree at position k (for all k) has no more that k+n vertices.
 * 2) No tree is homeomorphically embeddable into any tree following it in the sequence.

Adam P. Goucher has shown following properties of this function:


 * 1) tree(n) has growing level \(\vartheta(\Omega^\omega)\)  in the fast-growing hierarchy.
 * 2) TREE(3) > treetree tree tree tree 8(7) (7) (7) (7) (7)

Also found larger lower bound for TREE(3).

Define \(tree_2(n)=tree^n(n)\) and \(tree_3(n)=tree_2^n(n)\). Then TREE(3) > tree\(_3\) (tree\(_2\) (tree(8))), where \(tree_3 (tree_2 (tree(8) ) ) \) is about treetree tree ... tree(n)... (n) (n) (n) with n exponents, where \(n = tree_2 (tree(8)) = tree^{tree(8)} (tree(8)) \).