Neural models optimized for tree-based problems are of great value in tasks like SQL query extraction and program synthesis. On sequence-structured data, transformers have been shown to learn relationships across arbitrary pairs of positions more reliably than recurrent models. Motivated by this property, we propose a method to extend transformers to tree-structured data, enabling sequence-to-tree, tree-to-sequence, and tree-to-tree mappings. Our approach abstracts the transformer's sinusoidal positional encodings, allowing us to instead use a novel positional encoding scheme to represent node positions within trees. We evaluated our model in tree-to-tree program translation and sequence-to-tree semantic parsing settings, achieving superior performance over both sequence-to-sequence transformers and state-of-the-art tree-based LSTMs on several datasets. In particular, our results include a 22% absolute increase in accuracy on a JavaScript to CoffeeScript translation dataset.