This paper studies lower bounds for arithmetic circuits computing (non-commutative) polynomials. Our conceptual contribution is an exact correspondence between circuits and weighted automata: algebraic branching programs are captured by weighted automata over words, and circuits with unique parse trees by weighted automata over trees.

The key notion for understanding the minimisation question of weighted automata is the Hankel matrix: the rank of the Hankel matrix of a word or tree series is exactly the size of the smallest weighted automaton recognising this series. For automata over words, the correspondence we establish allows us to rephrase Nisan's celebrated tight bounds for algebraic branching programs. We extend this result by considering automata over trees and obtain the first tight bounds for all circuits with unique parse trees.