Associativity conditions for linear quasigroups and equivalence relations on binary trees

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.