  It is well known that the Young lattice is the Bratelli diagram of the
symmetric groups expressing how irreducible representations restrict from $S_N$
to $S_{N-1}$. In 1988, Stanley discovered a similar lattice called the
Young-Fibonacci lattice which was realized as the Bratelli diagram of a family
of algebras by Okada in 1994. In this paper, we realize the Okada algebra and
its associated monoid using a labeled version of Temperley-Lieb arc-diagrams.
We prove in full generality that the dimension of the Okada algebra is $n!$. In
particular, we interpret a natural bijection between permutations and labeled
arc-diagrams as an instance of Fomin's Robinson-Schensted correspondence for
the Young-Fibonacci lattice. We prove that the Okada monoid is aperiodic and
describe its Green relations. Lifting those results to the algebra allows us to
construct a cellular basis of the Okada algebra. }
