  We present generalizations of the Newton-Raphson-Simpson method.
Specifically, for a positive integer $m$ and the sequence of coefficients of a
Taylor series of a function $f(z)$, we define an algorithm we denote by
NRS($m$) which is a way to evaluate, in our terminology, a sum of $m$ formal
zeros of $f(z)$. We prove that NRS(1) yields the familiar iterations of the
Newton-Raphson-Simpson method. We also prove that NRS($m$) is way to evaluate
certain $\mathscr{A}$-hypergeometric series defined by Sturmfels. In order to
define these algorithms, we make use of combinatorial objects which we call
trees with negative vertex degree.
