Trees and the bireflection property

The group of automorphisms of a tree (partially ordered set where the set of predecessors of an element is well ordered) with no infinite levels enjoys the property that every member is a product of two elements of order ≦2. It is shown that this property-called the bireflection property-fails for some trees having infinite levels. In fact, every subtree of a tree T has the the bireflection property if and only if the tree of all zero-one sequences of length ≦ω with finitely many ones is not embeddable in T.