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We describe a completely algebraic axiom system for intertwining operators ofvertex algebra modules, using algebraic flat connections, thus formulating theconcept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, wefurther prove that a vertex tensor category in the sense of Huang and Lepowskygives rise to a tree algebra over $\C$. We also show that the chiral WZW modelof a simply connected simple compact Lie group gives rise to a tree algebraover $\Q$.

Tree Algebras: An algebraic axiomatization of intertwining vertex operators