The discourse analysis task, which focuses on understanding the semantics of long text spans, has received increasing attention in recent years. As a critical component of discourse analysis, discourse relation recognition aims to identify the rhetorical relations between adjacent discourse units (e.g., clauses, sentences, and sentence groups), called arguments, in a document. Previous works focused on capturing the semantic interactions between arguments to recognize their discourse relations, ignoring important textual information in the surrounding contexts. However, in many cases, more than capturing semantic interactions from the texts of the two arguments are needed to identify their rhetorical relations, requiring mining more contextual clues. In this paper, we propose a method to convert the RST-style discourse trees in the training set into dependency-based trees and train a contextual evidence selector on these transformed structures. In this way, the selector can learn the ability to automatically pick critical textual information from the context (i.e., as evidence) for arguments to assist in discriminating their relations. Then we encode the arguments concatenated with corresponding evidence to obtain the enhanced argument representations. Finally, we combine original and enhanced argument representations to recognize their relations. In addition, we introduce auxiliary tasks to guide the training of the evidence selector to strengthen its selection ability. The experimental results on the Chinese CDTB dataset show that our method outperforms several state-of-the-art baselines in both micro and macro F1 scores.