I have a continuous and finite tree graph $C$ with a finite set of points $P$ ($P$ contains all the vertices in $C$ and additional points). I wish to construct a discrete tree $D$ which preserves the distances in $C$, i.e., each point $p_i \in C$ is represented by a vertex $d_i \in D$, and for every $p_1,p_2,p_3 \in C$, s.t. $dist(p_1,p_2) > dist(p_1,p_3)$, it follows that $dist(d_1,d_2) > dist(d_1,d_3)$. How can I construct the tree assuming the distances between points in $P$ can be irrational?

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