XPath 1.0 nodes and axes

XPath 1.0 partitions the nodes in an XML document into 4 types (ignoring comments, processing instructions and namespaces):

• root - a document has exactly one of these; no parent; exactly one 'element' child
• element - parent must be 'element' or the 'root'; zero or more children of types 'attribute', 'element' or 'text'
• text - parent must be 'element'; no children
• attribute - parent must be 'element'; no children

XPath specifies a vocabulary of 12 'axes' - ancestor, descendant, preceding, following, preceding-sibling, following-sibling, self, ancestor-or-self, descendant-or-self, parent, child and attribute. There is a denotation function from axes and (context) nodes in a document to sets of nodes in the same document, defined as follows:

• [[self]]^D,n = {n}
• [[child]]^D,n = the set of non-attribute nodes in D which n immediately dominates
• [[attribute]]^D,n = the set of attribute nodes in D which n immediately dominates
• [[descendant]]^D,n = M ∪ U_m∈M [[descendant]]^D,m, where M = [[child]]^D,n
• [[descendant-or-self]]^D,n = [[descendant]]^D,n ∪ [[self]]^D,n
• [[parent]]^D,n = {m} where m is the parent of n
• [[ancestor]]^D,n = M ∪ U_m∈M [[ancestor]]^D,m, where M = [[parent]]^D,n
• [[ancestor-or-self]]^D,n = [[ancestor]]^D,n ∪ [[self]]^D,n
• [[preceding]]^D,n = the set of non-attribute nodes in D which precede n (i.e. finish before n starts)
• [[following]]^D,n = the set of non-attribute nodes in D which follow n (i.e. that start after n finishes)
• [[preceding-sibling]]^D,n = [[preceding]]^D,n ∩ {n' | ∃ m∈[[parent]]^D,n, n'∈[[child]]^D,m}
• [[following-sibling]]^D,n = [[following]]^D,n ∩ {n' | ∃ m∈[[parent]]^D,n, n'∈[[child]]^D,m}

You can read the XPath 1.0 reccomendation here.