  The idea of "vertex at the infinity" naturally appears when studying
indecomposable injective representations of tree quivers. In this paper we
formalize this behavior and find the structure of all the indecomposable
injective representations of a tree quiver of size an arbitrary cardinal
$\kappa$. As a consequence the structure of injective representations of
noetherian $\kappa$-trees is completely determined. In the second part we will
consider the problem whether arbitrary trees are source injective
representation quivers or not.
