TREE sequence

The TREE sequence is a very fast-growing function arising out of.

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. Currently, the limit of BEAF is \(f_{\varphi(2,0,0)}(n)\), so TREE(3) probably surpasses many or all of Bowers' numbers, including meameamealokkapoowa oompa. It is certainly much smaller than Rayo's number, however.

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