which $p$ and $\theta$ can be adapted simultaneously would be superior to both.

Discussion and Summary {#sec:summ}
======================

Our code for the approximate computation of mutual long-range forces between $N$ bodies extends the traditional Barnes & Hut [@BH] tree code by including cell-cell interactions, similar to fast multipole methods (FMM). However, unlike most implementations of FMM, our code is optimized for comparably low-accuracy, which is sufficient in stellar dynamical applications ([§\[sec:fmm\]]{}).

As a unique feature, our code exploits the mutual character of gravity: both nodes of any interaction are sink and source simultaneously. This results in exact conservation of Newton’s third law and substantially reduces the computational effort, but requires a novel tree-walking algorithm ([§\[sec:walk\]]{}) which preserves the natural symmetry of each interaction. Note that the generic algorithm given in [§\[sec:walk\]]{} is not restricted to long-range force approximations, but can be used for any task that incorporates mutuality, for instance neighbor and collision-partner searching.

Complexity {#sec:scal}
----------

The new code requires ${\cal O}(N)$ or less operations ([Fig. \[fig:N\]]{}) for the approximate computation of the forces of $N$ mutually gravitating bodies. For stellar dynamicists, this is the first competitive code better than ${\cal
  O}(N\log N)$. A complexity of ${\cal O}(N)$ was 