within the node from the center of mass (bodies naturally have $r_{\rm max}=0$). We take $r_{\rm max}$ to be either the distance from the cell’s center of mass to its most distant corner [@SW], or [@Benz] $$\max_{{\rm child\;nodes}\;i}\{r_{i,\rm max}+|{{\mbox{\boldmath$z$}}}-{{\mbox{\boldmath$z$}}}_i|\},$$ whatever is smaller. Then $|{{\mbox{\boldmath$x$}}}-{{\mbox{\boldmath$y$}}}-{{\mbox{\boldmath$R$}}}|<\theta|{{\mbox{\boldmath$R$}}}|\;\forall\;{{\mbox{\boldmath$x$}}}\in{{\rm A}}\;{\rm
  and}\;{{\mbox{\boldmath$y$}}}\in{{\rm B}}$, i.e. the nodes are [*well-separated*]{}, if $$\label{well-separated}
  |{{\mbox{\boldmath$R$}}}| > r_{\rm A,crit} + r_{\rm B,crit}
  \quad{\rm with}\quad
  r_{\rm crit} = r_{\rm max} / \theta.$$ The [*tolerance parameter*]{} $\theta$ controls the accuracy of the approximation: for Newtonian forces, the error made in $d$ dimensions by the $p$th order expansion of the form [(\[local-expn\])]{} is ([equation [(\[error:acc\])]{}]{}) $$\begin{aligned}
 \label{error:rel}
  |{\cal R}_p({\mbox{\boldmath$\nabla$}}\Phi_{{{\rm B}}\to{{\rm A}}})| &\le&
  {(p+1)\theta^p\over(1-\theta)^2}\;{{{\sf M}^{}}_{{\rm B}}\over R^2}
\\   \label{error:abs} &\propto&
  {\theta^{p+2}\over(1-\theta)^2}\;r_{\rm B,max}^{d-2}
  \propto
  {\theta^{p+2}\over(1-\theta)^2}\;{{{\sf M}^{}}_{{\rm B}}}^{(d-2)/d}\end{aligned}$$ where ${{\sf M}^{}}_{{\rm B}}\propto r_{\rm B,max}^d$ has been assumed. Since ${{\sf M}^{}}_{{\rm B}}/R^2$ is, to lowest order, the acceleration due to the interaction, [equation [(\[error:rel\])]{}]{} tells us that with constant $\theta$, which is standard tree-code practice, the [*relative*]{} error introduced by every single interaction is approximately constant. [Equation [(\[error:abs\])]{}]{}, however, shows that for the most important case of $d=3$ this practice results in larger [*absolute*]{} errors for interactions with bigger and hence on average more massive cells. This implies 