  A groupoid identity is said to be linear of length $2k$ if the same $k$
variables appear on both sides of the identity exactly once. We classify and
count all varieties of groupoids defined by a single linear identity. For
$k=3$, there are 14 nontrivial varieties and they are in the most general
position with respect to inclusion. Hentzel et. al. showed that the linear
identity $(xy)z = y(zx)$ implies commutativity and associativity in all
products of at least 5 factors. We complete their project by showing that no
other linear identity of any length behaves this way, and by showing how the
identity $(xy)z = y(zx)$ affects products of fewer than 5 factors; we include
distinguishing examples produced by the finite model builder Mace4. The
corresponding combinatorial results for labelled binary trees are given. We
associate a certain wreath product with any linear identity. Questions about
linear groupoids can therefore be transferred to groups and attacked by
group-theoretical computational tools, e.g., GAP. Systematic notation and
diagrams for linear identities are devised. A short equational basis for
Boolean algebras involving the identity $(xy)z = y(zx)$ is presented, together
with a proof produced by the automated theorem prover Otter.
