We address the problem of any-code completion – generating a missing piece of source code in a given program without any restriction on the vocabulary or structure. We introduce a new approach to any-code completion that leverages the strict syntax of programming languages to model a code snippet as a tree – structural language modeling (SLM). SLM estimates the probability of the program's abstract syntax tree (AST) by decomposing it into a product of conditional probabilities over its nodes. We present a neural model that computes these conditional probabilities by considering all AST paths leading to a target node. Unlike previous techniques that have severely restricted the kinds of expressions that can be generated in this task, our approach can generate arbitrary code in any programming language. Our model significantly outperforms both seq2seq and a variety of structured approaches in generating Java and C# code. We make our code, datasets, and models publicly available.