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Multivariate techniques are implemented in order to build, study and renormalise branched zeta functions associated with rooted trees. For this purpose, we first prove algebraic results and develop analytic tools, which we then combine to study branched zeta functions. The algebraic aspects concern universal properties for locality algebraic structures; we branch/lift to trees operators on the decoration set, and factorise branched maps through words by means of universal properties for words. The analytic tools arise in the context of multivariate meromorphic germs of symbols with linear poles. The latter form a locality algebra on which we build various locality maps such as locality Rota-Baxter operators given by regularised sums and integrals. Using locality universal properties, we lift Rota-Baxter operators and branched sums to decorated rooted trees to build and study branched zeta functions associated with trees. These renormalised branched zeta functions are multiplicative on mutually independent trees.