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. Mathematician Harvey Friedman expanded on this by asking the question: what is the maximum length of such a sequence for given values of k?

The length of such a sequence 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 can be shown that TREE(n) grows faster than \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy.

Chris Bird in his work "Beyond Bird's Nested Arrays II" (page 10) determined the number of TREE (3) as:         TREE(3)> {3, 6, 3 [1 [1 ¬ 1,2] 2] 2}.

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 \(\psi(\Omega^\omega)\)  in the fast-growing hierarchy.
 * 2) TREE(3) > treetree tree tree tree 8(7) (7) (7) (7) (7)

Notice that Jonathan Bowers' BEAF goes only up to \(\varphi(2,0,0)\) in the fast-growing hierarchy, so TREE(3) should be certainly greater than meameamealokkapoowa oompa.