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

Associativity conditions for linear quasigroups and equivalence relations on binary trees