  Kurz et al. have recently shown that infinite $\lambda$-trees with finitely
many free variables modulo $\alpha$-equivalence form a final coalgebra for a
functor on the category of nominal sets. Here we investigate the rational
fixpoint of that functor. We prove that it is formed by all rational
$\lambda$-trees, i.e. those $\lambda$-trees which have only finitely many
subtrees (up to isomorphism). This yields a corecursion principle that allows
the definition of operations such as substitution on rational $\lambda$-trees.
