  Sparse residual tree (SRT) is an adaptive exploration method for multivariate
scattered data approximation. It leads to sparse and stable approximations in
areas where the data is sufficient or redundant, and points out the possible
local regions where data refinement is needed. Sparse residual forest (SRF) is
a combination of SRT predictors to further improve the approximation accuracy
and stability according to the error characteristics of SRTs. The hierarchical
parallel SRT algorithm is based on both tree decomposition and adaptive radial
basis function (RBF) explorations, whereby for each child a sparse and proper
RBF refinement is added to the approximation by minimizing the norm of the
residual inherited from its parent. The convergence results are established for
both SRTs and SRFs. The worst case time complexity of SRTs is
$\mathcal{O}(N\log_2N)$ for the initial work and $\mathcal{O}(\log_2N)$ for
each prediction, meanwhile, the worst case storage requirement is
$\mathcal{O}(N\log_2N)$, where the $N$ data points can be arbitrary
distributed. Numerical experiments are performed for several illustrative
examples.
