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In a spoken dialogue system, the major aim of spoken language understanding (SLU) is to detect the dialogue acts (DAs) of a speakers utterance. However, error-prone speech recognition often degrades the performance of the SLU. In this work, a DA detection approach using partial sentence trees (PSTs) and a latent dialogue act matrix (LDAM) is presented for SLU. For each input utterance with speech recognition errors, several partial sentences (PSs) derived from the recognized sentence can be obtained to construct a PST. A set of sentence grammar rules (GRs) is obtained for each PS using the Stanford parser. The relationship between the GRs and the DAs is modeled by an LDAM. Finally, the DA with the highest probability estimated from the speech recognition likelihood, the LDAM and the historical information is determined as the detected DA. In evaluation, compared to the slot-based method which achieved 48.1% detection accuracy, the proposed approach can achieve 84.3% accuracy.