Geodesic
Uniqueness of Steiner minimal trees on boundaries
Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340

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The following result is proved: there exists an open dense subset $U$ of $\mathbb R^{2n}$ such that each $P\in U$ (regarded as an enumerated subset of the standard Euclidean plane $\mathbb R^2$) is spanned by a unique Steiner minimal tree, that is, a shortest non-degenerate network. Several interesting consequences are also obtained: in particular, it is proved that each planar Steiner tree is planar equivalent to a Steiner minimal tree. Bibliography: 11 titles. 
@article{SM_2006_197_9_a3,
     author = {A. O. Ivanov and A. A. Tuzhilin},
     title = {Uniqueness of  {Steiner} minimal trees on  boundaries},
     journal = {Sbornik. Mathematics},
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     volume = {197},
     number = {9},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/}
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A. O. Ivanov; A. A. Tuzhilin. Uniqueness of  Steiner minimal trees on  boundaries. Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340. http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/