Trees and valuation rings

A subring B of a division algebra D is called a valuation ring of D if x is an element of B or x(-1) is an element of B holds for all nonzero x in D. The set B of all valuation rings of D is a partially ordered set with respe ct to inclusion, having D as its maximal element. As a graph B is a rooted tree (called the valuation tree of D), and in contrast to the commutative c ase, B may have finitely many but more than one vertices. This paper is mai nly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra D, and one main result h ere is a positive answer to this question where D can be chosen as a quater nion division algebra over a commutative field.