Geodesic
Path Reconstruction in Barning–Hall Tree
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 95-102 Cet article a éte moissonné depuis la source Math-Net.Ru

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In 1963 F. J. M. Barning and in 1970 A. Hall gave a systematic procedure to generate all primitive Pythagorean triples (PPTs) with help of multiplication of the minimal Pythagorean triple $[3,4,5]$ considered as a vector by matrix series where matrices are taken from a prescribed 3-set of unimodular matrices. This provides a structure of an infinite ternary rooted labeled tree. In this article we establish an algorithm that reconstructs a tree path leading from the root to the primitive Pythagorean triple of interest. Because a triple can lie deeply then efficiency of the algorithm is quite important. The established algorithm has polynomial time complexity with respect to the input length relating to a “size” of PPT.
Keywords: pythagorean triples, Barning–Hall tree, triple generators, path reconstruction, algorithm efficiency. 
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P. G. Emelyanov. Path Reconstruction in Barning–Hall Tree. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 12 (2012) no. 3, pp. 95-102. http://geodesic.mathdoc.fr/item/VNGU_2012_12_3_a7/

[1] Barning F. J. M., Over Pythagorese en Bijna-Pythagorese Driehoeken en een Generatieproces met Behulp van Unimodulaire Matrices (On Pythagorean and Quasi-Pythagorean Triangles and a Generation Process with the Help of Unimodular Matrices), Afdeling Zuivere Wiskunde ZW 1963-011, Mathematisch Centrum, Amsterdam, 1963 (In Dutch) | MR | Zbl

[2] Hall A., “Genealogy of Pythagorean Triads”, Mathematical Gazette, 54:390 (1970), 377–379 | DOI

[3] Romik D., “The Dynamics of Pythagorean Triples”, Trans. AMS, 360 (2008), 6045–6064 | DOI | MR | Zbl

[4] Kanga A. R., “The Family Tree of Pythagorean Triples”, Bulletin of the Institute for Mathematics and its Applications, 26:1–2 (1990), 15–17 | MR