<p>This paper is concerned with the error of the fast multipole method (FMM) for the linearized Poisson-Boltzmann equation in 2-D and 3-D. We first give sharp bounds for the truncation errors of the expansions used in the algorithm, and then derive the error bounds of the FMM. The error analysis results show that the accuracy of FMM is related to the centers of the multipole expansions and local expansions. The expansion centers default to the centroid of the cell (square or cube) in the tree structure, but this may not necessarily be the best choice. We design an efficient linear time algorithm to optimize the expansion centers of the cells in the tree structure. 2-D and 3-D numerical experiments show that, it only takes a few seconds to optimize the tree structure, and the accuracy of FMM is significantly improved.</p>