#### Formal languages and automata VII: Formal tree series (Part II)

AbstractThis is the eighth paper of a series of papers that will give a survey on several topics on formal languages and automata by using semirings, formal power series, matrices and xed point theory. The seventh paper of this series deals with tree (series) automata and systems of equations over tree . The main topics of the paper are the following. 1. Pushdown tree automata, whose behaviors are tree series over a semiring, and algebraic tree systems are equivalent; moreover, the class of algebraic tree series is characterized by algebraic tree series expressions (a Kleene result). 2. The class of recognizable tree series is closed under nondeterðministic simple recognizable tree series transductions.3. The families of recognizable tree series and of algebraic tree series are full abstract families of tree series (full AFTs). 4. The macro power series, a generalization of the indexed languages, and the algebraic power series are exactly the yields of algebraic tree series and of recognizable tree series, respectively; there is a Kleene result for macro power series; the yield of a full AFT is a full abstract family of power series.