  In this paper we generalize to non-uniform grids of quad-tree type the
Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput.,
2001), thus obtaining a truly two-dimensional non-oscillatory third order
reconstruction with a very compact stencil and that does not involve
mesh-dependent coefficients. This latter characteristic is quite valuable for
its use in h-adaptive numerical schemes, since in such schemes the coefficients
that depend on the disposition and sizes of the neighboring cells (and that are
present in many existing WENO-like reconstructions) would need to be recomputed
after every mesh adaption.
  In the second part of the paper we propose a third order h-adaptive scheme
with the above-mentioned reconstruction, an explicit third order TVD
Runge-Kutta scheme and the entropy production error indicator proposed by Puppo
and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on
the choice of the parameters controlling the mesh adaption, we demonstrate with
many numerical tests that the scheme can compute numerical solution whose error
decays as $\langle N\rangle^{-3}$, where $\langle N\rangle$ is the average
number of cells used during the computation, even in the presence of shock
waves, by making a very effective use of h-adaptivity and the proposed third
order reconstruction.
