  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.
