Exploding Tree Function

The  Exploding Tree Function is the fast-growing googological function.

Definition
We have the binary tree T with m left nodes and n right nodes, some constant p and two transformation rules:


 * Replace T with a right-branching chain of length (n+p);
 * For each of the (n+p) nodes in the new subtree, attach a right-branching chain of length (m-1) as its left child.

The function determines iff becomes a chain of right-branching nodes.

Values
Since writing trees in full is impractical, let f(xLnR,m) is the obtained by the full transform tree with left-branching "length" of x and right-branching "length" of n and constant is m. Then E(n) = f(nL0R,n), a left-branching tree with n nodes and the constant n.

\( E(0) = 0 \) \( E(1) = 1 \) \( E(2) = 2 \) \( E(3) = 6561 \) \( E(4) > \lbrace 4,5,3,2 \rbrace \) \( E(5) > \lbrace 5,6,4,3 \rbrace \)

In general:

\( E(n) > \lbrace n,n+1,n-1,n-2 \rbrace \)

Comparison with other functions
E(n) surpasses cg(n) at n=4, since lower bounds for E(n) is upper bounds for cg(n) (see Bird's Proof).

E(n) might be upper bounded by \lbrace n,3n,n-1,n-2 \rbrace and hence by \(f_{\omega^2}(n)\) in the fast-growing hierarchy.