Not to be confused with TREE sequence, or Friedman's finite trees.

The Exploding Tree Function is a fast-growing googological function.[1]


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 computation process of the function terminates iff the tree becomes a chain of right-branching nodes.


Since writing trees in full is impractical, let f(xLnR,m) be the cardinality of 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.

\( E(0) = 0 \)
\( E(1) = 1 \)
\( E(2) = 2 \)
\( E(3) = 6,561 \)
\( E(4) > 4 \uparrow^{4 \uparrow\uparrow 4} 4 = \{4,4,\{4,4,2\}\} > \{4,2,1,2\} \)
\( E(5) > \{5,5,\{5,5,\{5,5,3\}\}\} > \{5,3,1,2\} \)
\( E(6) > \{6,6,\{6,6,\{6,6,\{6,6,4\}\}\}\} > \{6,4,1,2\} \)

In general, old members of this wiki accepted the estimation

\( E(n) > \{n,n-2,1,2\} \)

until 09/06/2021 as a fact,[2] but there is no source of a written proof.

Comparison with other functions

In the fast growing hierarchy, this wiki accepted the description unti 09/06/2021 as a fact that E(n) is comparable to \(f_{\omega+1}(n)\) and therefore it exhibits the expandal growth rate,[3] but there is no source of a written proof.


See also

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