TREE sequence

The TREE sequence is a very fast-growing function arising out of, devised by mathematical logician 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 famously very large; it has been proven that TREE(3) > (5) for any "reasonable amount" X.

TREE(n) grows faster than \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy, making it quite sizable even to a googologist &mdash; \(\vartheta(\Omega^\omega)\) describes the strength of BEAF linear array-arrays. It is more powerful than Kirby-Paris hydras and the Goodstein sequences, but weaker than subcubic graph numbers and Buchholz hydras.

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
Define \(\text{tree}(n)\), the weak tree function, as the length of the 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. In BEAF, \(\text{tree}(n) \approx \{X,X (1) 2\} \&\ n\).
 * 2) TREE(3) > treetree tree tree tree 8(7) (7) (7) (7) (7)

A larger lower bound has since been found. Define \(\text{tree}_2(n)=\text{tree}^n(n)\) and \(\text{tree}_3(n)=\text{tree}_2^n(n)\). Then \(\text{TREE}(3) > \text{tree}_3(\text{tree}_2(\text{tree}(8)))\). The latter expression is equal to treetree tree ... tree(n)... (n) (n) (n) with n layers, where \(n = \text{tree}^{\text{tree}(8)}(\text{tree}(8))\).

Values for \(\text{tree}(n)\)
It can be shown that \(\text{tree}(1) = 2, \text{tree}(2) = 5\) and \(\text{tree}(3) \geq 2^{18}-4\). \(\text{tree}(1)\) uses the sequence:

()

\(\text{tree}(2)\) is a bit larger, we have two longest sequences that go as follows:

(()) () () ()

Otherwise:

() ((())) (()) ()

Determining the exact value for \(\text{tree}(3)\) is much harder, since we have many long sequences and quite large number of them itself.

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 shall call a pair of corresponding parentheses a node, in deference to the original tree construction. Define a child of a node to be a node that is nested one level deep within the original node. For example, take the string {[][][]}; the children of the node represented by {} are the nodes represented by [], but not the nodes represented by.

Call a node deletable if it has fewer than two children. For example, in the string {[][][]}, the nodes are all deletable, as are the latter two [] nodes, but not the first [] or the {}.

We say a string A is reducible to a string B if A can be transformed into B by removing deletable nodes. A string A is reducible to a string B if and only if the tree represented by B is homeomorphically embeddable in the tree represented by A.

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 reducible to an earlier 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
Suppose we have a sequence of strings with the following properties:


 * 1) You may only use   and no other types of brackets.
 * 2) The first string has at most 1 + k pairs of brackets, the second string has at most 2 + k pairs of brackets, the third string has at most 3 + k pairs of brackets, etc.
 * 3) No string is carvable from a later string.

tree(k) (the weak tree function) is the maximal length of the sequence.