  This work is a contribution to the study of set of the representations of
integers in a rational base number system. This prefix-closed subset of the
free monoid is naturally represented as a highly non regular tree whose nodes
are the integers and whose subtrees are all distinct. With every node of that
tree is then associated a minimal infinite word. The main result is that a
sequential transducer which computes for all n the minimal word associated with
n+1 from the one associated with n, has essentially the same underlying graph
as the tree itself. These infinite words are then interpreted as
representations of real numbers; the difference between the numbers represented
by these two consecutive minimal words is the called the span of a node of the
tree. The preceding construction allows to characterise the topological closure
of the set of spans.
