To any path connected topological space , such that for all , are associated the following two sequences of integers: and . If is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that , for all , it is a classical result that the sequence of Betti numbers grows polynomially or exponentially, depending on whether the sequence is eventually zero or not. The purpose of this note is to prove, in both cases, that the Betti number is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space .