A fast parallel oct-tree code originally developed for three-dimensional N-body gravitational simulations was modi ed into (1) a fast N-vortex code for viscous and inviscid vortex ow computations using the regularized vortex particle method (VEM), and (2) a fast N-panel code for solving boundary integral equations in potential ow aerodynamics using the boundary element method (BEM). The core of the fast tree code remains essentially unchanged between the different application codes: gravitation, VEM, BEM, etc. Only the modules that actually encode the physical model are changed. Particular attention is given to controlling the error introduced by the use of multipole expansions to represent the eld produced by groups of elements, i.e., the tree code error. In particular, the acceptable error bound for use of any m ultipole expansion approximation is a run-time parameter. Program outputs include statistics on the errors
226with sample computational results. For the VEM method, a high order particle redistribution scheme has been incorporated, in an e cient w ay, i n to the parallel tree code. It is applied, if necessary, to ensure that the core overlapping condition remains satis ed in long time computations. In addition, two di erent relaxation schemes have also been incorporated and partially tested. Such s c hemes are applied, if necessary, to ensure that the particle representation of the vorticity eld remains a good representation of the true divergence free vorticity eld in long time computations.