  Stochastic Taylor expansions of the expectation of functionals applied to
diffusion processes which are solutions of stochastic differential equation
systems are introduced. Taylor formulas w.r.t. increments of the time are
presented for both, It\^o and Stratonovich stochastic differential equation
systems with multi-dimensional Wiener processes. Due to the very complex
formulas arising for higher order expansions, an advantageous graphical
representation by coloured trees is developed. The convergence of truncated
formulas is analyzed and estimates for the truncation error are calculated.
Finally, the stochastic Taylor formulas based on coloured trees turn out to be
a generalization of the deterministic Taylor formulas using plain trees as
recommended by Butcher for the solutions of ordinary differential equations.
