Summary: The motivation for considering positive additive functions on trees was a characterization of extended Dynkin graphs (see \textit{I.~Reiten} [Notices Am. Math. Soc. 44, No. 5, 546-556 (1997; Zbl 0940.16009)]) and applications of additive functions in representation theory (see \textit{H.~Lenzing} and \textit{I.~Reiten} [Colloq. Math. 82, No. 1, 85-103 (1999; Zbl 0984.16015)] and \textit{T.~Hübner} [Colloq. Math. 75, No. 2, 183-193 (1998; Zbl 0902.16012)]). We consider graphs equipped with integer-valued functions, i.e. valued graphs (see also \textit{V. Dlab} and \textit{C. M. Ringel} [Mem. Am. Math. Soc. 173 (1976; Zbl 0332.16015)]). Methods are given for constructing additive functions on valued trees (in particular on Euclidean graphs) and for characterizing their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except one (called the exceptional vertex). On (valued) trees (with fixed exceptional vertex) the almost additive functions are unique up to rational multiples. For valued trees a necessary and sufficient condition is given for the existence of positive almost additive functions.