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 for k = 3, and two approximation algorithms for general k, with ratios O({the square root of}n log k) and k+ε.
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