  In this paper, we consider a theoretical framework for comparing
branch-and-bound with classical lift-and-project hierarchies. We simplify our
analysis of streamlining the definition of branch-and-bound. We introduce
"skewed $k$-trees" which give a hierarchy of relaxations that is incomparable
to that of Sherali-Adams, and we show that it is much better for some
instances. We also give an example where lift-and-project does very well and
branch-and-bound does not. Finally, we study the set of branch-and-bound trees
of height at most $k$ and effectively "squeeze" their effectiveness between two
well-known lift-and-project procedures.
