On extensions of the Newton-Raphson iterative scheme to arbitrary orders

The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation $f(t)=0$ has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves.