Estimating the result sizes of XML queries is important in query optimization and is useful in providing a quick feedback about the queries. Existing works have focused on the selectivity estimation of XML queries without order-based axes. In this work, we develop a framework to estimate the result sizes of XPath expressions with order-based axes. We describe how the path and order information of XML elements can be captured and summarized in compact data structures. We also describe methods to estimate the selectivity of XPath queries. The results of extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness and accuracy of the proposed approach.