Weighted tree transducers have been proposed as useful formal models for representing syntactic natural language processing applications, but there has been little description of inference algorithms for these automata beyond formal foundations. We give a detailed description of algorithms for application of cascades of weighted tree transducers to weighted tree acceptors, connecting formal theory with actual practice. Additionally, we present novel on-the-fly variants of these algorithms, and compare their performance on a syntax machine translation cascade based on (Yamada and Knight, 2001). 1 Motivation Weighted finite-state transducers have found recent favor as models of natural language (Mohri, 1997). In order to make actual use of systems built with these formalisms we must first calculate the set of possible weighted outputs allowed by the transducer given some input, which we call forward application, or the set of possible weighted inputs given some output, which we call bac...