Geodesic
The Structure of Minimal Steiner Trees in the Neighborhoods of the Lunes of Their Edges
Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 353-370

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We give a complete description of small neighborhoods of the closures of lunes of the edges of Steiner minimal trees (Theorem 1.1); to this end, we prove a generalization of a stabilization theorem for embedded locally minimal trees [1]; the case of two such disjoint trees is considered (Theorem 2.2).
Keywords: Steiner minimal tree, locally minimal tree, lune of an edge of a tree, linear graph, shortest tree. 
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     author = {A. O. Ivanov and O. A. S'edina and A. A. Tuzhilin},
     title = {The {Structure} of {Minimal} {Steiner} {Trees} in the {Neighborhoods} of the {Lunes} of {Their} {Edges}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {353--370},
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A. O. Ivanov; O. A. S'edina; A. A. Tuzhilin. The Structure of Minimal Steiner Trees in the Neighborhoods of the Lunes of Their Edges. Matematičeskie zametki, Tome 91 (2012) no. 3, pp. 353-370. http://geodesic.mathdoc.fr/item/MZM_2012_91_3_a3/