Geodesic
The geometry of plane locally minimal binary trees
Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1271-1301

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An effective bound is obtained for the possible topology of a plane locally minimal binary tree in terms of its rotation number and the geometry of the boundary set. 
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     title = {The geometry of plane locally minimal binary trees},
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A. O. Ivanov. The geometry of plane locally minimal binary trees. Sbornik. Mathematics, Tome 186 (1995) no. 9, pp. 1271-1301. http://geodesic.mathdoc.fr/item/SM_1995_186_9_a2/