TREE sequence

The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other "large" combinatorial constants, such as Friedman's n(4), are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of A's is A(187196), and A is a version of Ackermann function: A(x) = 2↑↑...↑x with x-1 ↑s (Knuth up-arrows). Graham's number, for example, is approximately A64(4) which is much smaller than the lower bound AA(187196)(1).

TREE(k) is defined as the length of a longest sequence of k-labelled trees T1,...,Tm in which each Ti has at most i vertices, and no tree is homeomorphically embeddable into a later tree.