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Traversing the Nodes of a Tree
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There are several distinct ways of traversing, or sequencing, the nodes of an ordered tree. Three important ones are described in the literature: preorder, postorder, and inorder. They can be explained by means of the following recursive procedures (pseudocode again): operator preorder ( root ) ; emit root ; do for each child ( root ) left to right ; emit preorder ( child ( root ) ) ; end do ; end preorder ; operator postorder ( root ) ; do for each child ( root ) left to right ; emit postorder ( child ( root ) ) ; end do ; emit root ; end postorder ; operator inorder ( root ) ; emit inorder ( first child ( root ) ) ; emit root ; do for each child ( root ) left to right after the first ; emit inorder ( child ( root ) ) ; end do ; end inorder ; For example, consider the tree shown in Figure 14-1 once again. Assume it s ordered as that figure suggests. Then:
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CHAPTER 14 TREE-STRUCTURED DATA
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Preorder yields: A, B, C, D, F, E, G, H, I Postorder yields: C, F, D, E, B, H, I, G, A Inorder yields: C, B, F, D, E, A, H, G, I Actually, any ordered tree is equivalent in a specific sense to a certain ordered binary tree (see, e.g., Donald E. Knuth s book The Art of Computer Programming Volume I: Fundamental Algorithms, 3rd edition, Addison-Wesley, 1997, pages 334 335). We could make life a little simpler, therefore, by defining the foregoing operators in terms of ordered binary trees only: operator preorder ( root ) ; emit root ; emit preorder ( left child ( root ) ) ; emit preorder ( right child ( root ) ) ; end preorder ; operator postorder ( root ) ; emit postorder ( left child ( root ) ) ; emit postorder ( right child ( root ) ) ; emit root ; end postorder ; operator inorder ( root ) ; emit inorder ( left child ( root ) ) ; emit root ; emit inorder ( right child ( root ) ) ; end inorder ; As you can see, the simplified version of inorder in particular here is identical to the pseudocode procedure of the same name in the previous section, which explains why I used that name in that section.
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I ll now show how to do the generalized i.e., nonbinary version of preorder relationally. (I ll leave postorder and inorder as exercises. They re very similar, of course.) First, here in outline is an appropriate database design. I ve assumed for simplicity that the values in the nodes are integers. NODE { ID INTEGER, V INTEGER } KEY { ID } PC { PARENT INTEGER, FIRST_CHILD INTEGER } KEY { PARENT } KEY { FIRST_CHILD } LR { LEFT INTEGER, RIGHT INTEGER } KEY { LEFT } KEY { RIGHT } NODE { ID, V, FIRST_CHILD, NEXT_SIB } KEY { ID }
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CHAPTER 14 TREE-STRUCTURED DATA
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Explanation: NODE contains a tuple for each node in the tree. PC contains a node for each parent in the tree; PARENT gives the ID of that parent and FIRST_CHILD gives the ID of that parent s first child. LR contains a tuple for each child except the last for each parent in the tree; LEFT gives the ID of such a child and RIGHT gives the ID of the next child to the right. Node IDs are positive integers 1, 2, ... (as in the previous section). Observe that PARENT, FIRST_CHILD, LEFT, and RIGHT are never zero (there s no need with this design to use zero to mean some node doesn t exist). Note: For completeness, I ought really to state the integrity constraints that apply to relvars NODE, PC, and LR. I ve omitted those constraints here for space reasons, but you might like to try formulating them yourself before reading any further. By the same token, I ve also omitted the code for populating those relvars. Here now is the Tutorial D preorder code (it might look a little complicated, but it can easily be improved in a variety of ways). Assuming relvar NODE is nonempty (implying that the tree is nonempty), the Tutorial D expression PREORDER(1) will return the desired result as a relation with attributes P and V (where, as in the previous section, P = i corresponds to the value appearing in the ith position in the sequenced result). OPERATOR PREORDER ( M INTEGER ) RETURNS RELATION { P INTEGER, V INTEGER } ; BEGIN ; VAR OUTLIST RELATION { P INTEGER, V INTEGER } KEY { P } ; VAR TN TUPLE { ID INTEGER, V INTEGER } ; VAR TPC TUPLE { PARENT INTEGER, FIRST_CHILD INTEGER } ; VAR TLR TUPLE { LEFT INTEGER, RIGHT INTEGER } ; VAR X INTEGER ; /* OUTLIST tuple number */ X := ( IF M = 1 THEN 1 ELSE MAX ( OUTLIST, P ) + 1 END IF ) ; TN := TUPLE FROM ( NODE WHERE ID = M ) ; /* root */ INSERT OUTLIST RELATION { TUPLE { P X, V ( V FROM TN ) ) } } ; IF NOT ( IS_EMPTY ( PC WHERE PARENT = M ) ) THEN BEGIN ; TPC := TUPLE FROM ( PC WHERE PARENT = M ) ; INSERT OUTLIST PREORDER ( FIRST_CHILD FROM TPC ) ; IF NOT ( IS_EMPTY ( LR WHERE LEFT = ( FIRST_CHILD FROM TPC ) ) ) THEN BEGIN ; TLR := TUPLE FROM ( LR WHERE LEFT = ( FIRST_CHILD FROM TPC ) ) ) ;
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