$T_A$ += $T_0$; (all[body]{} children of $A$) {   evaluate $T_A$ at[body]{}’s position;   add to[body]{}’s potential and acceleration; } [**for**]{}(all[cell]{} children $C$ of $A$)   [Evaluate Gravity]{}($C$, $T_A$);

Thus, the coefficients of the Taylor-series that is eventually evaluated at some body’s position have been added up from all hierarchies of the tree and hence account for all interactions of all cells that contain the body.

Error Assessment {#sec:assess}
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Two types of force errors are involved in collisionless $N$-body simulations of stellar dynamics. One is the unavoidable error between the smooth force field of the underlying stellar system modeled and the forces [*estimated*]{} from the positions of $N$ bodies (which are sampled from this stellar system). This [ *estimation error*]{} can be reduced by increasing the number $N$ of bodies in conjunction with a careful [*softening*]{}: the Newtonian Greens function $g({{\mbox{\boldmath$x$}}})=G/|{{\mbox{\boldmath$x$}}}|$ is replaced with a non-singular function that approaches the Newtonian form for $|{{\mbox{\boldmath$x$}}}|$ larger than the softening length $\epsilon$. But at fixed $N$, it cannot be decreased below a certain optimum value [@Merr; @Db].

The other type of error is introduced by an approximate rather than exact computation of these estimated forces. While this [*approximation error*]{} can be reduced to (almost) 