User blog comment:PsiCubed2/My attempt for creating an ψ(ψᵢ(0))-level notation for ordinals/@comment-24920136-20170329070238/@comment-5529393-20170403005435

There is a natural ordering on Buchholz trees, where a subtree with a root labelled with n and children \(\alpha_1,\ldots,\alpha_n\) is equivalent to a collapsing function \(\psi_n (\alpha_1, \ldots, \alpha_n\).  In this ordering, +-0-1 is the \(\varepsilon_0\)th tree and I think +0-1-0-1 is the \(\zeta_0\)th tree.  If we change the rule for reducing a postive label to replicating the subtree n times rather than once, then we reduce to a fundamental sequence, so one can see that that the Buchholz hydra function for labels <= a will grow at the rate of \(F_{\psi_0(\varepsilon_{\Omega_{a+1}+1})}\) (as always, using the same fundamental sequences).  However, Buchholz uses a simplified rule where the subtree is only replicated once, yet somehow the function grows at a similar rate.  I'm not sure exactly why that is.

Buchholz's original paper is available here:  https://www.researchgate.net/publication/265713596_An_independence_result_for_P_1_1_-CABI