Geodesic
Universal boundary sets in the generalized Steiner problem
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 168-182

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It is proved that every simple binary tree is realized by a nongenerate minimal parametric net spanning the vertices of a regular simplex. 
@article{ZNSL_2001_279_a9,
     author = {G. A. Karpunin},
     title = {Universal boundary sets in the generalized {Steiner} problem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {168--182},
     publisher = {mathdoc},
     volume = {279},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a9/}
}
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G. A. Karpunin. Universal boundary sets in the generalized Steiner problem. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 6, Tome 279 (2001), pp. 168-182. http://geodesic.mathdoc.fr/item/ZNSL_2001_279_a9/