In the UI you posted it makes sense for the Tree or better the Panel not to have any scrollbars as the size of your tree is set to Full. If you use setSizeUndefined() instead and still don’t see scrollbars when the Tree expands over the size of the Panel it really is a bug.
SizeFull means the Tree will expand to the maximum possible size of its parent.
(I just tested it and yes using setSizeUndefined will cause scrollbars to appear as desired)
But! Tree can itself show scrollbars. In 7.3 release this functionality is broken. If I add Tree to VerticalLayout and then add VerticalLayout to Panel Tree show scrollbars normally. This bug in CSS:
I noticed this change as of Vaadin 7.3.3 and higher. Now if you setSizeFull on the Tree object then its size matches the alloted screen size and lacks scrollbars. My trees are wrapped in a Panel and I fixed it by removing setSizeFull method calls.