Enumerative Sequences of Leaves and Nodes in Rational Trees

We prove that any IN-rational sequence s = (s n ) n1 of nonnegative integers satisfying the Kraft strict inequality P n1 s n k \Gamman ! 1 is the enumerative sequence of leaves by height of a rational k-ary tree. We give an efficient algorithm to get a k-ary rational tree. Particular cases of this result had been previously proven. We give some partial results in the case of equality. Especially we solve this question when the associated sequence of internal nodes has a primitive linear representation