Locally Minimal Trees in $n$-Normed Spaces

The locally minimal trees in normed spaces $({\mathbb R}^2,\rho)$, where the unit circle $\Sigma=\{x\in{\mathbb R}^2\mid{\rho}(x)=1\}$ in the norm $\rho$ coincides with the regular $m$-gon ($m = 2n$) inscribed in the Euclidean unit circle $S^1$, are completely classified.