Some remarks on recognizable treeseries

Let K be a commutative ring, Σ a finite ranked alphabet and T _Σ the set of trees over Σ. We show that for any recognizable treeseries S:T → K having finite range and any subset A of K, the forest of all trees t ? T _Σ such that the corresponding coefficient (s, t) belongs to Ais recognizable.

In the case _Σ collapses to a monadic alphabet, we refind an analogous result concerning rational wordseries due to Schutzenberger [11] and Sontag [12]. Some applications are also given.