  Cambrian trees are oriented and labeled trees which fulfill local conditions
around each node generalizing the conditions for classical binary search trees.
Based on the bijective correspondence between signed permutations and leveled
Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday
and M. Ronco's algebra on binary trees. We describe combinatorially the
products and coproducts of both the Cambrian algebra and its dual in terms of
operations on Cambrian trees. We also define multiplicative bases of the
Cambrian algebra and study structural and combinatorial properties of their
indecomposable elements. Finally, we extend to the Cambrian setting different
algebras connected to binary trees, in particular S. Law and N. Reading's
Baxter Hopf algebra on quadrangulations and S. Giraudo's equivalent Hopf
algebra on twin binary trees, and F. Chapoton's Hopf algebra on all faces of
the associahedron.
