Dialogue state tracking is a key component of a task-oriented dialogue system. Recently, pretrained language models are widely used to track a dialogue state by generating or extracting values from a dialogue. These models have a shortcoming that they can explicitly encode contextual information of a dialogue. In this paper, we present a Syntactically ENriched Discourse Dialogue State Tracking model (SEND-DST) that fully leverages the discourse and syntax information of a dialogue. Our model consists of two parts: a dialogue encoding module and a slot-value extraction module. To provide a model with syntactic structures of utterances and the discourse flow of a dialogue, we devise a Syntactic Discourse (SD) graph using the dependency trees of the utterances. Experimental results show that SEND-DST achieves state-of-the-art results on multiple datasets, such as MultiWOZ 2.1, demonstrating its effectiveness.