  Given an arbitrary ordered pair of coprime integers (a,b) we obtain a pair of
identities of the Rogers--Ramanujan type. These identities have the same
product side as the (first) Andrews--Gordon identity for modulus 2ab\pm 1, but
an altogether different sum side, based on the representation of (a/b-1)^{\pm
1} as a continued fraction. Our proof, which relies on the Burge transform,
first establishes a binary tree of polynomial identities. Each identity in this
Burge tree settles a special case of Bressoud's generalized Borwein conjecture.
