  We prove the existence of self-similar expanding solutions of the curvature
flow on planar networks where the initial configuration is any number of
half-lines meeting at the origin. This generalizes recent work by Schn\"urer
and Schulze which treats the case of three half-lines. There are multiple
solutions, and these are parametrized by combinatorial objects, namely Steiner
trees with respect to a complete negatively curved metric on the unit ball
which span $k$ specified points on the boundary at infinity. We also provide a
sharp formulation of the regularity of these solutions at $t=0$.
