  A $k$-decision tree $t$ (or $k$-tree) is a recursive partition of a matrix
(2D-signal) into $k\geq 1$ block matrices (axis-parallel rectangles, leaves)
where each rectangle is assigned a real label. Its regression or classification
loss to a given matrix $D$ of $N$ entries (labels) is the sum of squared
differences over every label in $D$ and its assigned label by $t$. Given an
error parameter $\varepsilon\in(0,1)$, a $(k,\varepsilon)$-coreset $C$ of $D$
is a small summarization that provably approximates this loss to \emph{every}
such tree, up to a multiplicative factor of $1\pm\varepsilon$. In particular,
the optimal $k$-tree of $C$ is a $(1+\varepsilon)$-approximation to the optimal
$k$-tree of $D$.
  We provide the first algorithm that outputs such a $(k,\varepsilon)$-coreset
for \emph{every} such matrix $D$. The size $|C|$ of the coreset is polynomial
in $k\log(N)/\varepsilon$, and its construction takes $O(Nk)$ time. This is by
forging a link between decision trees from machine learning -- to partition
trees in computational geometry.
  Experimental results on \texttt{sklearn} and \texttt{lightGBM} show that
applying our coresets on real-world data-sets boosts the computation time of
random forests and their parameter tuning by up to x$10$, while keeping similar
accuracy. Full open source code is provided.
