The paper presents a modified binary tree in the fast multipole method (FMM) included into the modified parametric integral equations system (PIES), called the fast PIES, in solving potential 2D boundary value problems with complex shapes. The modified binary tree proposed in this paper is built based on a one-dimensional reference system contrary to a quad-tree (based on a two-dimensional reference system) which is applied in the fast multipole boundary element method (FM-BEM). Application of the proposed tree allows reducing the number of numerical computations performed during its construction and fast multipole calculations in the fast PIES. The proposed modification of the tree in the fast PIES allows obtaining accurate solutions in engineering problems with complex shapes on a standard personal computer in a short time.