TREE sequence

The TREE sequence is a fast-growing function arising out of, devised by mathematical logician Harvey Friedman. Friedman proved that the function eventually dominates all recursive functions provably total in the system \(\text{ACA}_0\)+\(\Pi_2^1\)-\(\text{BI}\).

The smallest non-trivial (that is to say, significantly larger than number we encounter in everyday experience) member of the sequence is the famously large TREE(3) (or TREE[3] with respect to the original convention), notable because it is a number that appears in serious mathematics that is larger than Graham's number.

Definition
Suppose we have a sequence of k-s T1, T2 ... with the following properties: 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?
 * 1) Each tree Ti has at most i vertices.
 * 2) No tree is inf preserving and label preserving embeddable into any tree following it in the 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 vastly exceeds Graham's number and nn(5)(5). Chris Bird claimed that \(\text{TREE}(3) > \{3, 6, 3 [1 [1 \neg 1,2] 2] 2\}\), using his array notation.

TREE(n) is strongly believed to grow at least as fast as \(f_{\vartheta(\Omega^\omega\omega)}(n)\) in the fast-growing hierarchy, making it quite sizable even to a googologist, but the estimation has not been verified. If it is correct, then it is more powerful than Kirby-Paris hydras and the Goodstein sequences, but weaker than subcubic graph numbers and Buchholz hydras. For a detail on the issue on the estimation, see also this.

Weak tree function
Define \(\text{tree}(n)\), the weak tree function, as the length of the longest sequence of 1-labelled trees such that: Adam P. Goucher has shown the following properties of this function: 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 \(n = \text{tree}^{\text{tree}(8)}(\text{tree}(8))\).
 * 1) Every tree at position k (for all k) has no more than k + n vertices.
 * 2) No tree is homeomorphically embeddable into any tree following it in the sequence.
 * 1) tree(n) has a "growth rate" comparable to that of \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy.
 * 2) TREE(3) > treetree tree tree tree 8(7) (7) (7) (7) (7)

Values for \(\text{tree}(n)\)
It can be shown that \(\text{tree}(1) = 2, \text{tree}(2) = 5\) and \(\text{tree}(3) \geq 844424930131960\). \(\text{tree}(1)\) 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 there are a lot of sequences to check, and each of these is very long.

Friedman has defined an FFF(k) function, which is equal to tree(k+1).

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. In the string, the   node is deletable. (Note: is this correct?)

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

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