  Using the method of spectral decimation and a modified version of Kirchhoff's
Matrix-Tree Theorem, a closed form solution to the number of spanning trees on
approximating graphs to a fully symmetric self-similar structure on a finitely
ramified fractal is given in Theorem \ref{thm:maintheoremfull}. We show how
spectral decimation implies the existence of the asymptotic complexity constant
and obtain some bounds for it. Examples calculated include the Sierpinski
Gasket, a non post critically finite analog of the Sierpinski Gasket, the
Diamond fractal, and the Hexagasket. For each example, the asymptotic
complexity constant is found.
