The implementation of the convolution method for the numerical solution of backward stochastic differential equations (BSDEs) presented in Hyndman and Oyono Ngou (2013) uses a uniform space grid. Locally, this approach produces a truncation error, a space discretization error and an additional extrapolation error. Even if the extrapolation error is convergent in time, the resulting absolute error may be high at the boundaries of the uniform space grid. In order to solve this problem, we propose a tree-like grid for the space discretization which suppresses the extrapolation error leading to a globally convergent numerical solution for the BSDE. On this alternative grid the conditional expectations involved in the BSDE time discretization are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm as in the initial implementation. The method is then extended to reflected BSDEs and numerical results demonstrating convergence are presented.