  The purpose of this paper is twofold. First we answer to a question asked by
Steingrimsson and Williams about certain permutation tableaux: we construct a
bijection between binary trees and the so-called Catalan tableaux. These
tableaux are certain Ferrers (or Young) diagrams filled with some 0's and 1's,
satisfying a certain hook condition, and are enumerated by the Catalan numbers.
They form a subclass of the permutation tableaux, enumerated by n!, introduced
by Postnikov in his study of totally non negative Grassmannians and networks.
Secondly we relate this new Catalan bijection with the totally asymmetric
exclusion process (TASEP), a very rich and well studied 1D gas model in
statistical mechanics of nonequilibrium systems. We continue some combinatorial
understanding of that model, in the spirit of works by Shapiro, Zeilberger and
more recently by Brak, Essam, Rechnitzer, Corteel, Williams, Duchi and
Schaeffer. Emphasis is made on the non-classical notion of canopy of a binary
tree, analog of the classical up-down sequence of a permutation.
