Functional code clone detection is important for software maintenance. In recent years, deep learning techniques are introduced to improve the performance of functional code clone detectors. By representing each code snippet as a vector containing its program semantics, syntactically dissimilar functional clones are allowed to be detected. However, existing deep learning-based approaches attach too much importance to code feature learning, hoping to project all recognizable knowledge of a code snippet into a single vector. We argue that these deep learning-based approaches can be enhanced by considering the characteristics of syntactic code clone detection, where we need to compare the contents of the source code (e.g., intersection of tokens, similar flow graphs, and similar subtrees) to obtain code clones. In this paper, we propose a novel deep learning-based approach named DSFM, which incorporates comparisons between code snippets for detecting functional code clones. Specifically, we improve the typical deep clone detectors with deep subtree interactions that compare every two subtrees extracted abstract syntax trees (ASTs) of two code snippets, thereby introducing more fine-grained semantic similarity. By conducting extensive experiments on three widely-used datasets, GCJ, OJClone, and BigCloneBench, we demonstrate the great potential of deep subtree interactions in code clone detection task. The proposed DSFM outperforms the state-of-the-art approaches, including two traditional approaches, two unsupervised and four supervised deep learning-based baselines.