More precisely, the tree decomposition has the same nodes and the same branching structure as the derivation tree, and each bag in the tree decomposition contains all the vertices that appear in the right-hand side of the rule.
the derivation tree must satisfy the three properties in the definition of a tree decomposition: Vertex cover: Each vertex in G must be generated by some rule in the HRG derivation tree t. Edge cover: Each edge (u, v) in G must also be generated by some rule r, and therefore both u and v appear in the right-hand side of r, and in the node for r in t. Running intersection: Any vertex v of G is first generated by some rule r of t. Vertex v can only appear in another rule r if it is connected to the lefthand side nonterminal of r , and therefore appears in the parent of r in t. all occurrences of v form a connected subtree of t. See also Lautemann (1988) for further discussion.