XML and other semi-structured data may have partially specified or missing schema information, motivating the use of a structural summary which can be automatically computed from the data. These summaries also serve as indices for evaluating the complex path expressions common to XML and semi-structured query languages. However, to answer all path queries accurately, summaries must encode information about long, seldom-queried paths, leading to increased size and complexity with little added value. We introduce the A(k)-indices, a family of approximate structural summaries. They are based on the concept of k-bisimilarity, in which nodes are grouped based on local structure, i.e., the incoming paths of length up to k. The parameter k thus smoothly varies the level of detail (and accuracy) of the A(k)-index. For small values of k, the size of the index is substantially reduced. While smaller, the A(k) index is approximate, and we describe techniques for efficiently extracting exact answ...