  We initiate the study of affine actions of groups on $\Lambda$-trees for a
general ordered abelian group $\Lambda$; these are actions by dilations rather
than isometries. This gives a common generalisation of isometric action on a
$\Lambda$-tree, and affine action on an $\R$-tree as studied by I. Liousse. The
duality between based length functions and actions on $\Lambda$-trees is
generalised to this setting. We are led to consider a new class of groups:
those that admit a free affine action on a $\Lambda$-tree for some $\Lambda$.
Examples of such groups are presented, including soluble Baumslag-Solitar
groups and the discrete Heisenberg group.
