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 first significantly large member of the sequence is the famous TREE[3] (sometimes written as TREE(3)), 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 some 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, and is 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. If this belief is indeed correct, then the TREE function is more powerful than Kirby-Paris hydras and Goodstein sequences, but weaker than subcubic graph numbers and Buchholz hydras. For a look into one of the issues of estimating high growth rates, check this blog post by Jason.

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 than k + n vertices.
 * 2) No tree is homeomorphically embeddable into any tree following it in the sequence.

Issues on Analysis
Since TREE and tree are quite difficult to evaluate, there are so many wrong or proofless statements about them. For example, several videos and articles explain how large TREE[3] is by "comparing" to many other known numbers without sources. Those estimations might be fortunately true, but the truth is irrelevant to the fact that they have not been proved.

For instance, Adam P. Goucher stated the following properties of this function: His claims lacked proof until July 24, 2020, when, in response to this issue, he supplied a proof of the latter in an edit of the same blog post. Later, on the same day, he provided proofs in another blog post of both claims. However, Googology Wiki user P進大好きbot claims these proofs are based on common mistakes.
 * 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)

Although there are few known results, i.e. proved statements, of lower bounds of tree, a famous Japanese mathematician T. Kihara verified that tree(4) is greater than Graham's number. He introduced a hierarchy \(\text{tree}_n\) of large functions indexed by ordinals \(n \leq \omega + 1\) associated to a hierarchy of binary trees equipped with a fixed coding of a sequence of natural numbers, and precisely estimated the hierarchy in terms of fast-growing hierarchy. The significance of his works is that he explicitly estimated the value of the combinatoric large functions, while many googologists roughly talks about "the corresponding ordinals" without proofs. The ability of trees to express ordinals does not ensure the comparison between \(\text{tree}\) and the fast-growing hierarchy, and hence many arguments based on this wrong deduction are meaningless.

Further, Kihara developed the method to precisely estimate the hierarchy \(\text{tree}_n\) extended to ordinals \(n < \Gamma_0\) associated to a hierarchy of ternary trees equipped with a fixed coding of Veblen hierachy, and verified \(\textrm{tree}(4) > f_{\varepsilon_0}(G)\) and \(\text{tree}(5) > f_{\Gamma_0}(G)\) with respect to the canonical system of fundamental sequences, where \(G\) denotes Graham's number. Unlike "usual" arguments in this community based on wrong deductions like "the fast-growing hierarchy depends only on ordinals" or "the existence of the correspondence to ordinals automatically ensures the comparability to the fast-growing hierarchy", the result explicitly estimates the lowerbounds of the specific values of tree function.

A larger lower bound of TREE[3] by tree than Adam P. Goucher's statement without a proof has since been found. Define \(\text{tree}_2(n)=\text{tree}^n(n)\) and \(\text{tree}_3(n)=\text{tree}_2^n(n)\). (Note that it differs from the hierarchy \(\text{tree}_n\) in the previous paragraph.) 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 \(\text{tree}(1) = 2, \text{tree}(2) = 5\), \(\text{tree}(3) \geq 844424930131960\), and \(\text{tree}(4) > \text{Graham's number}\). \(\text{tree}(1)\) 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).

Alternative notations
(This alternative has yet to be formally verified.)

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.

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 a function which should correspond with the original definition of TREE[k] assuming this notation was derived correctly. 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.