  This article is motivated by the objective of providing a new analytically
tractable and fully frequentist framework to characterize and implement
regression trees while also allowing a multivariate (potentially high
dimensional) response. The connection to regression trees is made by a high
dimensional model with dynamic mean vectors over multi-dimensional change axes.
Our theoretical analysis is carried out under a single two dimensional change
point setting. An optimal rate of convergence of the proposed estimator is
obtained, which in turn allows existence of limiting distributions.
Distributional behavior of change point estimates are split into two distinct
regimes, the limiting distributions under each regime is then characterized, in
turn allowing construction of asymptotically valid confidence intervals for
$2d$-location of change. All results are obtained under a high dimensional
scaling $s\log^2 p=o(T_wT_h),$ where $p$ is the response dimension, $s$ is a
sparsity parameter, and $T_w,T_h$ are sampling periods along change axes. We
characterize full regression trees by defining a multiple multi-dimensional
change point model. Natural extensions of the single $2d$-change point
estimation methodology are provided. Two applications, first on segmentation of
{\it Infra-red astronomy satellite (IRAS)} data and second to segmentation of
digital images are provided. Methodology and theoretical results are supported
with monte-carlo simulations.
