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 more compact 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:

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

Suppose we have a string A. We define a process called carving, which is doing any of the following actions:


 * Inside a string, all empty pairs  can be removed.
 * Example:
 * If we have a string  where   is any type of parenthesis and   are valid strings, we can replace it with   or   or   or ...
 * Example:
 * If we have a string  where   is any type of parenthesis, we can replace it with.
 * If we have a string  where   and   are two types of parenthesis (not necessarily distinct) and   is any string of parentheses.

A string A is embedded in another string B if we can repeatedly remove matching pairs in string B to form string A, removing only the innermost and outermost pairs and any stray pairs left outside. Here is an example of this "carving" process, with the removed pairs highlighted:

 ( [({[]})]{} )  - outermost [({[]})] {}  - stray pair  [ ({[]}) ]  - outermost ({[]})  - stray pair  ( {[]} )  - outermost {[ ]} - innermost {[]}

So  is embedded in.

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