Multi-point variants of the Newton-Raphson-Simpson method arising from organizing a formal zero according to a function $\varphi$

Fix an integer $L\ge 1$, and a function $\varphi \text{\hspace{0.17em}⁣}:{\mathbb{Z}}_{\ge 1}\to [0,L]\cap \mathbb{Z}$ with ${\varphi }^{-1}(\{0\})=\{1\}$. We define a multi-point variant of the Newton-Raphson-Simpson, which we call the max-phi method, as follows. We use $\varphi$ to define the iteration number of a rooted plane tree. Then we construct formal series that are weighted generating functions of rooted plane trees with iteration number at most $N$. Finally we use these formulas to define the max-phi method applied to an arbitrary $L$-differentiable function.