  We define a graded multiplication on the vector space of essential paths on a
graph $G$ (a tree) and show that it is associative. In most interesting
applications, this tree is an ADE Dynkin diagram. The vector space of length
preserving endomorphisms of essential paths has a grading obtained from the
length of paths and possesses several interesting bialgebra structures. One of
these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of
quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct
gives a filtrated convolution product on the dual vector space. Another
bialgebra structure is obtained by replacing this filtered convolution product
by a graded associative product.It can be obtained from the former by
projection on a subspace of maximal grade, but it is interesting to define it
directly, without using the DTA. What is obtained is a weak bialgebra, not a
weak Hopf algebra.
