  We characterise the bracketing identities satisfied by linear quasigroups
with the help of certain equivalence relations on binary trees that are based
on the left and right depths of the leaves modulo some integers. The numbers of
equivalence classes of $n$-leaf binary trees are variants of the Catalan
numbers, and they form the associative spectrum (a kind of measure of
non-associativity) of a quasigroup.
