  We study combinatorial and order theoretic structures arising from the
fragment of combinatory logic spanned by the basic combinator ${\bf M}$. This
basic combinator, named as the Mockingbird by Smullyan, is defined by the
rewrite rule ${\bf M} x_1 \to x_1 x_1$. We prove that the reflexive and
transitive closure of this rewrite relation is a partial order on terms on
${\bf M}$ and that all connected components of its rewrite graph are Hasse
diagrams of lattices. This last result is based on the introduction of lattices
on some forests. We enumerate the elements, the edges of the Hasse diagrams,
and the intervals of these lattices with the help of formal power series on
terms and on forests.
