  We establish a framework for cellularity of algebras related to the Jones
basic construction. Our framework allows a uniform proof of cellularity of
Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras,
partition algebras, and others. Our cellular bases are labeled by paths on
certain branching diagrams rather than by tangles. Moreover, for the class of
algebras that we study, we show that the cellular structures are compatible
with restriction and induction of modules. Applied to cyclotomic BMW algebras,
our method allows a new a shorter proof of the finite spanning result and
isomorphism with cyclotomic Kauffman tangle algebras.
