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We consider K-semialgebras for a commutative semiring K that are at the same time @S-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear K-@S-semialgebra. Examples of such algebras include the algebras of @S-tree series over an alphabet A with coefficients in K, and the algebra of all rational tree series. We show that for many commutative semirings K, the rational @S-tree series over A with coefficients in K form the free multi-linear iterative K-@S-semialgebra on A.