  This paper is concerned with near-optimal approximation of a given function
$f \in L_2([0,1])$ with elements of a polynomially enriched wavelet frame, a
so-called quarklet frame. Inspired by $hp$-approximation techniques of Binev,
we use the underlying tree structure of the frame elements to derive an
adaptive algorithm that, under standard assumptions concerning the local
errors, can be used to create approximations with an error close to the best
tree approximation error for a given cardinality. We support our findings by
numerical experiments demonstrating that this approach can be used to achieve
inverse-exponential convergence rates.
