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A particle-cluster treecode based on barycentric Lagrange interpolation is presented for fast summation of hydrodynamic interactions through general Rotne-Prager-Yamakawa tensor in 3D. The interpolation nodes are taken to be Chebyshev points of the 2nd kind in each cluster. The barycentric Lagrange interpolation is scaleinvariant that promotes the treecode’s efficiency. Numerical results show that the treecode CPU time scales like O (NlogN), where N is the number of beads in the system. The kernel-independent treecode is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.