# How Big would “Graham's Tree” be?

What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?

• What is "tree(3)"? – Arthur May 21 '17 at 12:26
• googology.wikia.com/wiki/TREE_sequence – Goodwin Lu May 21 '17 at 12:27
• By the way, "every current named number except infinity" makes no sense. Infinity is not a number – Arthur May 21 '17 at 12:31
• ah, alright. Some consider it to be. – Goodwin Lu May 22 '17 at 18:54
• Rayo's number is by design almost certainly larger than anything along these lines you could write down. – Mark S. May 22 '17 at 19:15

No. If you replaced all the $3$'s in the construction of Graham's number with $\operatorname{TREE}(3)$, the resulting number would be smaller than $g_{\operatorname{TREE(3)}}$ where $g_n$ denotes the $n$th number in Graham's sequence with $g_{64}$ being Graham's number. This is much much smaller than $\operatorname{TREE}(4)$, for example.
• Certainly smaller than $TREE(g_{TREE(3) + 64})$ – Francisco José Letterio Jan 13 '18 at 4:09
• Though $\operatorname{TREE}(4)$ is already a good enough upper bound. – Simply Beautiful Art Jan 13 '18 at 13:25