Quadratic forms and their Berggren trees

An old result of Berggren's says that there exist three 3×3 matrices N1,N2,N3 with the following remarkable property: Start with (3,4,5) or (4,3,5) and multiply N1,N2, or N3 by it in any order any number of times. This yields another primitive Pythagorean triple (x,y,z), that is, a triple of positive integers without common factor satisfying x2+y2−z2=0. Furthermore, every primitive Pythagorean triple can be obtained uniquely this way. In other words, all primitive Pythagorean triples can be given a tree-like structure with each edge representing a multiplication by Nj. In this paper, we present a geometric algorithm for producing such trees that is applicable to any integral quadratic form. Although this algorithm does not always yield a tree, we find a few other trees arising from different quadratic forms.