  We define a method which produces explicit cellular bases for algebras
obtained via a Jones basic construction. For the class of algebras in question,
our method gives formulas for generic Murphy--type cellular bases indexed by
paths on branching diagrams and compatible with restriction and induction on
cell modules. The construction given here allows for a uniform combinatorial
treatment of cellular bases and representations of the Brauer,
Birman-Murakami-Wenzl, Temperley-Lieb, and partition algebras, among others.
