Geodesic
Geometry of convex polygons and locally minimal binary trees spanning these polygons
Sbornik. Mathematics, Tome 190 (1999) no. 1, pp. 71-110

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In previous works the authors have obtained an effective classification of planar locally minimal binary trees with convex boundaries. The main aim of the present paper is to find more subtle restrictions on the possible structure of such trees in terms of the geometry of the given boundary set. Special attention is given to the case of quasiregular boundaries (that is, boundaries that are sufficiently close to regular ones in a certain sense). In particular, a series of quasiregular boundaries that cannot be spanned by a locally minimal binary tree is constructed. 
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     title = {Geometry of convex polygons and locally minimal binary trees spanning these polygons},
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A. O. Ivanov; A. A. Tuzhilin. Geometry of convex polygons and locally minimal binary trees spanning these polygons. Sbornik. Mathematics, Tome 190 (1999) no. 1, pp. 71-110. http://geodesic.mathdoc.fr/item/SM_1999_190_1_a2/