  We study piece-wise constant signals corrupted by additive Gaussian noise
over a $d$-dimensional lattice. Data of this form naturally arise in a host of
applications, and the tasks of signal detection or testing, de-noising and
estimation have been studied extensively in the statistical and signal
processing literature. In this paper we consider instead the problem of
partition recovery, i.e.~of estimating the partition of the lattice induced by
the constancy regions of the unknown signal, using the
computationally-efficient dyadic classification and regression tree (DCART)
methodology proposed by \citep{donoho1997cart}. We prove that, under
appropriate regularity conditions on the shape of the partition elements, a
DCART-based procedure consistently estimates the underlying partition at a rate
of order $\sigma^2 k^* \log (N)/\kappa^2$, where $k^*$ is the minimal number of
rectangular sub-graphs obtained using recursive dyadic partitions supporting
the signal partition, $\sigma^2$ is the noise variance, $\kappa$ is the minimal
magnitude of the signal difference among contiguous elements of the partition
and $N$ is the size of the lattice. Furthermore, under stronger assumptions,
our method attains a sharper estimation error of order
$\sigma^2\log(N)/\kappa^2$, independent of $k^*$, which we show to be minimax
rate optimal. Our theoretical guarantees further extend to the partition
estimator based on the optimal regression tree estimator (ORT) of
\cite{chatterjee2019adaptive} and to the one obtained through an NP-hard
exhaustive search method. We corroborate our theoretical findings and the
effectiveness of DCART for partition recovery in simulations.
