User blog comment:Mh314159/Help me understand a natural number recursion/@comment-35470197-20191014142325

That function employs the 2-adic encoding \begin{eqnarray*} \mathbb{N} \times \mathbb{N} & \to & \mathbb{N} \\ (a,b) & \mapsto & 2^a(2b+1)-1, \end{eqnarray*} which is known to be bijective. By the bijectivity, it allows us to define a multivariable function in terms of a function with fewer variables. Then the resulting system will be complicated, and hence is not good to study. If you want to study numerical coding of a computable large function, it is better to study an array system such as primitive sequence system.

By the way, the estimation of TREE is strange, because there are few known effective upperbounds of TREE. I guess that the original author misunderstood TREE as something like the ordinal \(\psi(\Omega^{\omega})\) in FGH.