The Rank of a Formal Tree Power Series

We prove that a tree series S: TΣ → K (K a field) is recognizable iff dim Vs < ∞ iff dim Fs < ∞, where Vs (resp. Fs) is the subspace if KPΣ (resp. KTΣ) generated by the vectors t-1 S=Στϵ PΣ (S, tτ)τ, tϵ τ)t, τϵPΣ), and where TΣ is the set of all trees over Σ and PΣ is the free monoid of all pruned trees over Σ. We finally give a Myhill-like criterion for tree-recognizability.