2. Tree Traversal

What I've just called ``scanning through'' a tree is actually called traversing a tree.

General Definition: to traverse a data structure is to process, however you like, every node in the data structure exactly once.

Note: You may ``pass through'' a node as many times as you like but you must only process the node once.

E.g. we can talk about ``traversing a list'', which means going through the list and processing every node once. We had a special name for this: map.

For a specific data structure, we talk about the different orders in which it might be traversed. For a list there are two common traversal orders: first-to-last (the most common) and last-to-first.

The general recursive pattern for traversing a (non-empty) binary tree is this: At node N you must do these three things:

(L) recursively traverse its left subtree. When this step is finished you are back at N again.
(R) recursively traverse its right subtree. When this step is finished you are back at N again.
(N) Actually process N itself.

We may do these things in any order and still have a legitimate traversal. If we do (L) before (R), we call it left-to-right traversal, otherwise we call it right-to-left traversal.

Pre-Order Traversal: do (N) before anything else.
Post-Order Traversal: do (N) after everything else.
In-Order, or Infix Order, Traversal: do (N) in between the two subtrees.

E.g. let us look at what order the nodes will get processed given this tree:

Using Left-To-Right traversal:

For example, suppose we were writing out the nodes of the tree:

        void print_tree(binary_tree *t)
        {
          if (! is_empty(t)) {
            print_tree(t->left);     /* L */
            print_tree(t->right);    /* R */
            printf("%d\n",t->value); /* N */
          }
        }

The preceding diagrams show us the order in which the nodes will get written:

Pre-Order: A-B-D-C-E-F
In-Order: B-D-A-E-C-F
Post-Order: D-B-E-F-C-A

Breadth First Traversal

These three are certainly not the only possible traversal orders. Another very natural traversal order is ``level by level'' - the root is processed first, all its children are processed next, then all of their children, etc. down to the bottom level. This is called breadth first traversal. In the above example, it would process nodes in the order: A-B-C-D-E-F. It is not difficult to write a breadth-first traversal, but is not quite as simple as the traversal orders just described.