Tree Lattices and the Neumann-Zagier conjecture

A tree with integer weights on its vertices presents an integral lattice called a tree lattice. We prove that if a tree lattice is positive definite, then one of the vertex elements is irreducible, i.e. determines a facet of its Voronoi polyhedron. As a corollary, we settle a 1985 conjecture of Neumann and Zagier: if a tree lattice is isometric to the lattice of integer points in Euclidean space, then the tree contains a vertex of weight one.