value of $N$ will depend on the tolerance parameter $\theta$ and the distribution of bodies.

Another way of estimating the scaling of the computational costs with $N$ is similar to the FMM approach [@FMMa; @FMMb]: On each level $l$ of the tree there are $\sim8^l$ cells of mass ${{\sf M}^{}}\sim8^{-l}$, i.e. $n({{\sf M}^{}})\propto{\sf
  M}^{-2}$. Each cell has $\propto\theta^{-3}$ interactions, and thus $$\label{scaling}
        N_I \propto \int { d{{\sf M}^{}}\over{\sf M}^{2}\theta^3}.$$ Hence, for $\theta=\,$const, $N_I\propto1/{\sf M}_{\rm min}\propto N$, while a shallower scaling results if $\theta({{\sf M}^{}})$ increases towards smaller masses. Empirically we find for $N>30000$ that the CPU time used by the force computation alone (without tree building) is very well fit by the power-law $\propto N^{0.929\pm0.001}$.

Comparison with Other Methods used in Stellar Dynamics {#sec:comp}
------------------------------------------------------

For various computational techniques used in stellar dynamics, [Figure \[fig:var\]]{} plots $N$ versus the CPU time consumption normalized by $N$, both on logarithmic scales. An ordinary direct method running on a general-purpose computer is slower than our code for any $N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}100$. The [GRAPE-5]{} system [@GRAPE] obtains a $\sim100$ times higher performance by wiring elementary gravity into special-purpose hardware and, for $N{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}10^4$, is faster than any other method.

The new code presented here is the fastest 