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Let $T$ be a $\R$-tree in the boundary of the Outer Space CV$_N$, with denseorbits. The $Q$-index of $T$ is defined by means of the dual lamination of $T$.It is a generalisation of the Euler-Poincar\'e index of a foliation on asurface. We prove that the $Q$-index of $T$ is bounded above by $2N-2$, and westudy the case of equality. The main tool is to develop the Rips Machine inorder to deal with systems of isometries on compact $\R$-trees. Combining ourresults on the $\CQ$-index with results on the classical geometric index of atree, we obtain a beginning of classification of trees. As a consequence, wegive a classification of iwip outer automorphisms of the free group, bydiscussing the properties of their attracting and repelling trees.

Rips Induction: Index of the dual lamination of an $\R$-tree