typedef struct node {
    struct node *child1;
    struct node *child2;
    char *data;
} node;

• It will often be binary like that.
• … but could be $m$-ary in general.
• And might be a binary search tree (everything in left smaller; in right larger).
• But might not be: we could just store whatever data in whatever child we want.
• For now, let's imagine it's just some (rooted) binary tree:

• … and that we have a pointer to the root node.

• If we did, we would probably have to process each element at some point.
• You have probably written this more times than you can count:

  for (int i=0; i<length; i++) {
      process(array[i]);
  }

• process() could be any operation: print the contents, add up the values, modify the data,…
• We don't care what operation it is, just that we had to touch every element.

• If we are given a pointer to the root node, how do we find every element so we can process it?
• Is there a better or worse way to do that?

procedure preorder_traverse(root)
    if root is not null
        process(root.data)
        preorder_traverse(root.child1)
        preorder_traverse(root.child2)

• If we did this on the above tree, we would visit the nodes in this order: $$a, b, d, g, h, c, e, f, i, j$$
• This is called a preorder traversal because we visit the node itself before visiting the children.
• Obviously, if we had an $m$-ary tree, we could visit each of the $m$ children recursively.
• We did get at all of the data in the tree, which was the goal.

• We will visit the node after its children.
• Like this:

  procedure postorder_traverse(root)
      if root is not null
          postorder_traverse(root.child1)
          postorder_traverse(root.child2)
          process(root.data)

• Then we get the data in a different order. In the example: $$g,h,d,b,e,i,j,f,c,a$$
• If we just stored our data randomly in the tree, the order probably doesn't matter: one order might be as good as another.
• Either way, we get at all of the data.

procedure inorder_traverse(root)
    if root is not null
        inorder_traverse(root.child1)
        process(root.data)
        inorder_traverse(root.child2)

• An inorder traversal.
• Just another order of the data.
• With our example tree: $$g,d,h,b,a,e,c,i,f,j$$

• … then an inorder traversal gets the data in a very nice order:

  apple, banana, cherry, date, fig, grape, honeydew

• It visited the elements in sorted order.

• Rules of a BST: everything in the left subtree is smaller; everything in the right subtree is bigger.
• So we visit the smaller things first, then the root node, then the larger things.
• We could easily prove by induction (on the number of nodes) that this gets us everything in sorted order.

• We get these items out in this order:

  apple, banana, cherry, date, eggplant, fig, grape, honeydew

• Usually we don't care: we mostly want to do inorder traversal on BSTs.
• But if we had to define inorder traversal for $m$-ary trees, I would visit child 1, then the root, then children 2 to $m$.

• Vertices are operators and numbers.
• The children of operators are the values they're operating on.
• Numbers are leaves.
• For the above expression, we get this:

• We needed to know the order of operations rules to build the tree correctly.
• A compiler builds basically this tree out of your code as it compiles it.

procedure evaluate(expr)
    if expr.is_an_operator
        operand1 = evaluate(expr.leftchild)
        operand2 = evaluate(expr.rightchild)
        return expr.operator applied to operand1 and operand2
    else if expr.is_a_number
        return expr.value

• … if we print a “(” first and “)” last around each operator, so we preserve the order of operations.
• For our sample tree, the output would be: $$((2-1)-(3+(4\times 2)))$$
• Some extra parens, but basically the same thing we started with.

• Hmmm… doesn't seem useful.
• But if we spell things a little differently, it makes sense.
• Turn “+” into “ADD(”, etc.
• Add a “,” between and “)” after each operator traversal.
• Then we get this:

  SUB(SUB(2, 1), ADD(3, MULT(4, 2)))

• We turned the infix operator notation into prefix notation for function calls.

• On our example tree, we get:

  2 1 - 3 4 2 × + -

• Also doesn't seem very useful at first glance.
• But it's the postfix notation needed by a stack-based calculator/processor.
• If you don't know what that is, don't worry about it.

• That's one of the reasons a compiler has to build that tree.
• The processor doesn't want to values in the infix order you write in your code.
• But the compiler can build the tree and traverse it to generate the assembly code needed.