TREE sequence

The TREE sequence is a very fast-growing function arising out of, devised by Harvey Friedman.

Definition
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.

states that such a sequence 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.

Explanation
Trees are tricky to visualize without drawing them out, so we shall devise a simpler way of representing them. Consider a language which has various kinds of parentheses such as,  ,   etc. Parentheses always come in pairs and can nest within each other. Within a larger node, they may be concatenated. For example, the following strings are valid in this language:

[] ([]) {[]} [{}{()[]}]

A string A is a substring of another string B if B is of the form "XAY," where X and Y are arbitrary and possibly empty strings. Informally, this means that A can be found inside B with no modifications. For example,  and   are substrings of the fourth string above. is not a substring, since it doesn't appear without modifications.

However, despite that trees is similar to strings, it is not the same. For trees, we can modify the larger tree such that remove neighboring vertices to obtain a smaller tree. For example,  is homeomorphically to   since we can remove  part from the second tree to obtain the first tree.

With all this in mind, we can create an equivalent definition of TREE(k). Suppose we have a sequence of strings with the following properties:


 * 1) You may only use k types of brackets.
 * 2) The first string has at most one pair of brackets, the second string has at most two pairs of brackets, the third string has at most three pairs of brackets, etc.
 * 3) No string is a substring of a later string.

TREE(k) is the maximal length of the sequence.

For k = 1, the optimal sequence has only one member:.

For k = 2, the optimal sequence has only two members:, then  , then.

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 \(n = tree_2 (tree(8)) = tree^{tree(8)} (tree(8)) \).