Chapter4b: Trees and tableaux : the Loday-Ronco algebra of binary trees

In 1998 Loday and Ronco defined a structure of graded Hopf algebra on theset of binary trees (enumerated by Catalan numbers). The construction involved the graded Hopf algebra on permutations defined by Malvenuto and Reutenauer in 1995. We will recall the notion of Catalan alternative tableaux (related to the TASEP model in physics), in bijection with binary trees and show their role in the definition of the product of binary trees in the Loday-Ronco Hopf algebra. A product can be defined directly on these tableaux which reflect very simply the Loday-Ronco product