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-1S=ΣτϵPΣ (S, tτ)τ, tϵτ)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.