the latter results in higher accuracy at the same efficiency, see Appendix B for more details.

If an interaction is executed by direct summation, the Taylor coefficients of the interacting cell(s) are not affected, but the coefficients ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{0}}$ and ${\tilde{{\mbox{\boldmath$\sf C$}}}{}^{1}}$ of the bodies within the cell(s) are accumulated.

The Algorithm {#sec:alg}
=============

Tree Building and Preparation {#sec:build}
-----------------------------

In the first stage, a hierarchical tree of cubic cells is build, as described in [@BH], albeit cells containing $s$ or less bodies are not divided. Next, the tree is linked such that every cell holds the number of cell children[^2], a pointer to its first cell child, as well as the number of body children, of all body descendants, and a pointer to the first body child. Since cell as well as body children are contiguous in memory (to arrange this we actually use tiny copies of the bodies, called ‘souls’, that only hold a pointer to their body and its mass, position, acceleration and potential) these data allow fast and easy access to all cell and body children of any given cell and to all bodies contained in it.

Next, the masses ${{\sf M}^{}}$, centers of mass ${{\mbox{\boldmath$z$}}}$, 