| Nonlinear piecewise polynomial (spline) approximation is important from many prospectives and has various applications. We will focus our attention on nonlinear n-term approximation from sequences of hierarchical spline bases generated by multilevel triangulations over polygonal domains in R^2. In contrast to the wavelet case, these are highly nonlinear approximation methods from redundant systems with a great deal of flexibility. It will be shown that the rates of nonlinear n-term spline approximation are governed by certain smoothness spaces, called B-spaces. Unlike the commonly used Besov spaces, the B-spaces allow us to characterize all rates of approximation, which gives more complete results in the isotropic case as well. The emphasis will be placed on algorithms for nonlinear n-term approximation from the scaling functions of a (spline) multiresolution analysis in L_p and in the uniform norm, which both capture the rates of the best n-term approximation and provide the basis for numerical implementation. |