  We prove that if a metric space $X$ has Nagata dimension zero with constant
$c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz
equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$,
that is, if $X$ is an ultrametric space. This yields a new proof of a result of
Chan, Xia, Konjevod and Richa. Moreover, as an application, we also obtain
quantitative versions of certain metric embedding and Lipschitz extension
results of Lang and Schlichenmaier. Finally, we prove a variant of our main
theorem for $0$-hyperbolic proper metric spaces. This generalizes a result of
Gupta.
