XML queries typically specify patterns of selection predicates on multiple elements that have some specified tree structured relationships. The primitive tree structured relationships are parent-child and ancestor-descendant, and finding all occurrences of these relationships in an XML database is a core operation for XML query processing. In this paper, we develop two families of structural join algorithms for this task: tree-merge and stack-tree. The tree-merge algorithms are a natural extension of traditional merge joins and the recently proposed multi-predicate merge joins, while the stack-tree algorithms have no counterpart in traditional relational join processing. We present experimental results on a range of data and queries using the TIMBER native XML query engine built on top of SHORE. We show that while, in some cases, tree-merge algorithms can have performance comparable to stack-tree algorithms, in many cases they are considerably worse. This behavior is explained by analytical results that demonstrate that, on sorted inputs, the stack-tree algorithms have worst-case I/O and CPU complexities linear in the sum of the sizes of inputs and output, while the tree-merge algorithms do not have the same guarantee.