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This paper connects the classes of weighted alternating finite automata(WAFA), weighted finite tree automata (WFTA), and polynomial automata (PA).  First, we investigate the use of trees in the run semantics for weightedalternating automata and prove that the behavior of a weighted alternatingautomaton can be characterized as the composition of the behavior of a weightedfinite tree automaton and a specific tree homomorphism, if weights are takenfrom a commutative semiring.  Based on this, we give a Nivat-like characterization for weighted alternatingautomata. Moreover, we show that the class of series recognized by weightedalternating automata is closed under inverses of homomorphisms, but not underhomomorphisms. Additionally, we give a logical characterization of weightedalternating automata, which uses weighted MSO logic for trees. Finally, weinvestigate the strong connection between weighted alternating automata andpolynomial automata. We prove: A weighted language is recognized by a weightedalternating automaton iff its reversal in recognized by a polynomial automaton.Using the corresponding result for polynomial automata, we are able to provethat the ZERONESS problem for weighted alternating automata with weights takenfrom the rational numbers decidable.

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings