Derivative coordinates for analytic tree fractals and fractal engineering

We introduce an alternative coordinate system based on derivative polar and spherical coordinate functions and construct a root-to-canopy analytic formulation for tree fractals. We develop smooth tree fractals and demonstrate the equivalence of their canopies with iterative straight lined tree fractals. We then consider implementation and application of the analytic formulation from a computational perspective. Finally we formulate the basis for concatenation and composition of fractal trees as a basis for fractal engineering of which we provide some examples.