Given a set of k-colored points in the plane, we consider the problem of
finding k trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
k=1, this is the well-known Euclidean Steiner tree problem. For general k,
a kρ-approximation algorithm is known, where ρ≤1.21 is the
Steiner ratio.
We present a PTAS for k=2, a (5/3+ε)-approximation algorithm
for k=3, and two approximation algorithms for general~k, with ratios
O(nlogk) and k+ε