  Multivariate renormalisation techniques are implemented in order to build,
study and then renormalise at the poles, branched zeta functions associated
with 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, some of which had been discussed in previous work; we "branch/
lift" to trees operators acting on the decoration set of trees, and factorise
branched maps through words by means of universal properties for words which we
prove in the locality setup. The analytic tools are multivariate meromorphic
germs of pseudodifferential symbols with linear poles which generalise the
meromorphic germs of functions with linear poles studied in previous work.
Multivariate meromorphic germs of pseudodifferential symbols form a locality
algebra on which we build various locality maps in the framework of locality
structures. We first show that the finite part at infinity defines a locality
character from the latter symbol valued meromorphic germs to the scalar valued
ones. We further equip the locality algebra of germs of pseudodifferential
symbols with locality Rota-Baxter operators given by regularised sums and
integrals. By means of the universal properties in the framework of locality
structures we can lift Rota-Baxter operators to trees, and use the lifted
discrete sums in order to build and study renormalised branched zeta values
associated with trees. By construction these renormalised branched zeta values
factorise on mutually independent (for the locality relation) trees.
