The Inverse Berggren Tree Problem

Berggren's three matrices generate the tree of primitive Pythagorean triples and preserve the quadratic form x₁² + x₂² − x₃². Cha, Nguyen and Tauber (2018) built such trees for other integral ternary forms Q of signature (2,1). We ask whether the matrices alone determine Q. Main results: Recovery criterion (Theorem 3.1). A set of isometries of Q determines Q up to scale exactly when no nonzero vector is sent to plus or minus itself by all of them. Equivalently, the group they generate is not virtually abelian; for integral isometries, it acts irreducibly. The proof is elementary linear algebra and applies to arbitrary real groups. Berggren trees (Theorem 3.2, Proposition 3.3, Remark 3.5). Any two generators determine Q in Berggren's tree and in each of the five trees of Cha, Nguyen and Tauber. The definition of a Berggren tree alone does not force this: the generators A and A² give a Berggren forest for x₁² + x₂² − x₃² whose two generators both yield edges but do not determine Q. Roots at a rational null line (Theorem 3.6). For a primitive null vector P with divisor G, the minimum of Q on (P^⊥ ∩ ℤ³) \ ℤP is S = |det(2J)|/(2G²). An explicit congruence, Gm ≡ −τ (mod S), decides which minimal vectors e + mP are roots. N-cusps (Theorems 3.7–3.8, Corollary 3.9). For Q = ax₁² + bx₁x₂ + cx₂² − Nx₃² with D = b² − 4ac < 0, all data at the null vectors (p, q, 1) are explicit. A minimal root exists there if and only if gcd(G, S) divides D/G, and in particular whenever D is cube-free. Recovery by integral reflections (Theorem 3.12, Proposition 3.13). Minimal roots at two or three distinct rational null lines yield three integral reflections that determine Q. The paper also relates the reflections in the trees of Cha, Nguyen and Tauber to minimal roots (Remark 3.17), and states an open question for trees produced by their construction.