  We prove that if a metric space $X$ has Nagata dimension zero with constant
$c$, then there exist a dense subset $Z\subset X$ and a weighted tree $Y$ such
that $Z$ and $Y$ are $8c$-bilipschitz equivalent. In particular, if $X$ is a
finite metric space and $c$ denotes its Nagata constant, then there exists a
weighted tree $T=(X, E, \omega)$ such that $X$ and $T$ are $8c$-bilipschitz
equivalent if $T$ is equipped with the shortest-path metric. We show that for
finite ultrametric spaces the constant $8$ is the best possible. This implies
that a result of Gupta on removing Steiner points from weighted trees is
optimal in general. Furthermore, as a corollary of our main result, we find an
explicit version of a Lipschitz extension theorem due to Lang and
Schlichenmaier in the special case when the source space is of Nagata dimension
zero.
