  We introduce a smooth projective variety $T_{d,n}$ which compactifies the
space of configurations of $n$ distinct points on affine $d$-space modulo
translation and homothety. The points in the boundary correspond to $n$-pointed
stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$,
are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is
isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety
$T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we
prove, the boundary is a smooth normal crossings divisor whose components are
products of $T_{d,i}$ for $i < n$, it has an inductive construction analogous
to but differing from Keel's for $\bar{M}_{0,n}$ which can be used to describe
its Chow groups, Chow motive and Poincar\'e polynomials, generalizing
\cite{Keel,Man:GF}. We give a presentation of the Chow rings of $T_{d,n}$,
exhibit explicit dual bases for the dimension 1 and codimension 1 cycles. The
variety $T_{d,n}$ is embedded in the Fulton-MacPherson spaces $X[n]$ for
\textit{any} smooth variety $X$ and we use this connection in a number of ways.
For example, to give a family of ample divisors on $T_{d,n}$ and to give an
inductive presentation of the Chow groups and the Chow motive of $X[n]$
analogous to Keel's presentation for $\bar{M}_{0,n}$, solving a problem posed
by Fulton and MacPherson.
