  This paper is concerned with model order reduction of parametric Partial
Differential Equations (PDEs) using tree-based library approximations.
Classical approaches are formulated for PDEs on Hilbert spaces and involve one
single linear space to approximate the set of PDE solutions. Here, we develop
reduced models relying on a collection of linear or nonlinear approximation
spaces called a library, and which can also be formulated on general metric
spaces. To build the spaces of the library, we rely on greedy algorithms
involving different splitting strategies which lead to a hierarchical
tree-based representation. We illustrate through numerical examples that the
proposed strategies have a much wider range of applicability in terms of the
parametric PDEs that can successfully be addressed. While the classical
approach is very efficient for elliptic problems with strong coercivity, we
show that the tree-based library approaches can deal with diffusion problems
with weak coercivity, convection-diffusion problems, and with
transport-dominated PDEs posed on general metric spaces such as the
$L^2$-Wasserstein space.
