Aspect Sentiment Triplet Extraction (ASTE) aims at extracting triplets from a given sentence, where each triplet includes an aspect, its sentiment polarity, and a corresponding opinion explaining the polarity. Existing methods are poor at detecting complicated relations between aspects and opinions as well as classifying multiple sentiment polarities in a sentence. Detecting unclear boundaries of multi-word aspects and opinions is also a challenge. In this paper, we propose a Multi-Task Dual-Tree Network (MTDTN) to address these issues. We employ a constituency tree and a modified dependency tree in two sub-tasks of Aspect Opinion Co-Extraction (AOCE) and ASTE, respectively. To enhance the information interaction between the two sub-tasks, we further design a Transition-Based Inference Strategy (TBIS) that transfers the boundary information from tags of AOCE to ASTE through a transition matrix. Extensive experiments are conducted on four popular datasets, and the results show the effectiveness of our model.