Geodesic
Minimal Networks on the Regular $n$-Dimensional Simplex
Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 854-865.

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The paper gives the proof of the following fact: all simple, i.e., having no nodes of degree 2, trees that span the vertices of the regular $n$-dimensional simplex can be realized as nondegenerate minimal parametric networks. 
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G. A. Karpunin. Minimal Networks on the Regular $n$-Dimensional Simplex. Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 854-865. http://geodesic.mathdoc.fr/item/MZM_2001_69_6_a4/

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