  This work presents a method to adaptively refine reduced-order models \emph{a
posteriori} without requiring additional full-order-model solves. The technique
is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis
space online by `splitting' a given basis vector into several vectors with
disjoint support. The splitting scheme is defined by a tree structure
constructed offline via recursive $k$-means clustering of the state variables
using snapshot data. The method identifies the vectors to split online using a
dual-weighted-residual approach that aims to reduce error in an output quantity
of interest. The resulting method generates a hierarchy of subspaces online
without requiring large-scale operations or full-order-model solves. Further,
it enables the reduced-order model to satisfy \emph{any prescribed error
tolerance} regardless of its original fidelity, as a completely refined
reduced-order model is mathematically equivalent to the original full-order
model. Experiments on a parameterized inviscid Burgers equation highlight the
ability of the method to capture phenomena (e.g., moving shocks) not contained
in the span of the original reduced basis.
