In computer science, tree traversal is the process of visiting each node in a tree data structure. Tree traversal, also called walking the tree, provides for sequential processing of each node in what is, by nature, a non-sequential data structure. Such traversals are classified by the order in which the nodes are visited.

The following algorithms are described for a binary tree, but they may be generalized to other trees.

Traversal methods

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Say we have a tree node structure which contains a value value and references left and right to its two children. Then we can write the following function:

pre-order (prefix) traversal

visit(node)
    print node.value
    if node.left  != null then visit(node.left)
    if node.right != null then visit(node.right)

This recursive algorithm prints the values in the tree in pre-order. In pre-order, each node is visited before any of its children. Similarly, if the print statement were last, each node would be visited after all of its children, and the values would be printed in post-order. In both cases, values in the left subtree are printed before values in the right subtree.

post-order (postfix) traversal

visit(node)
    if node.left  != null then visit(node.left)
    if node.right != null then visit(node.right)
    print node.value

in-order (infix) traversal

visit(node)
    if node.left  != null then visit(node.left)
    print node.value
    if node.right != null then visit(node.right)

An in-order traversal, as above, visits each node between the nodes in its left subtree and the nodes in its right subtree. This is a particularly common way of traversing a binary search tree, because it gives the values in increasing order.

To see why this is the case, note that if n is a node in a binary search tree, then everything in n's left subtree is less than n, and everything in n's right subtree is greater than or equal to n. Thus, if we visit the left subtree in order, using a recursive call, and then visit n, and then visit the right subtree in order, we have visited the entire subtree rooted at n in order. We can assume the recursive calls correctly visit the subtrees in order using the mathematical principle of structural induction.

If instead we visit the right subtree, then the current node, then the left subtree, this produces the opposite order as in-order traversal, sometimes called a reverse in-order or reverse-order traversal.

level order traversal

visitlevel(S)
  T = Set()
  for each node in S   
    print node.Value
    if (node->left != null)
       T.insert(node->left)
    if (node->right != null)
       T.insert(node->right)
  if ( T.empty() != true)
    visitlevel(T)
visit(root)
  S = Set()
  S.insert(root)
  visitlevel(S)

This prints the nodes in the order of their depth from the root. See the iterative version below.

Examples

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In this binary tree, Preorder (NLR) traversal yields: A, H, G, I, F, E, B, C, D Postorder (LRN) traversal yields: G, F, E, I, H, D, C, B, A In-order (LNR) traversal yields: G, H, F, I, E, A, B, D, C Level-order traversal yields: A, H, B, G, I, C, F, E, D

Functional traversal

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We could perform the same traversals in a functional language like Haskell using code like this:

data Tree a = Nil | Node (Tree a) a (Tree a)
preorder Nil = *
preorder (Node left x right) = * ++ (preorder left) ++ (preorder right)
postorder Nil = *
postorder (Node left x right) = (postorder left) ++ (postorder right) ++ *
inorder Nil = *
inorder (Node left x right) = (inorder left) ++ * ++ (inorder right)

Iterative traversal

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All of the above recursive algorithms use stack space proportional to the depth of the tree. If we store on each node a reference to its parent, then we can implement all of these traversals in-place using a simple iterative algorithm. The parent references occupy much more space, however; this is only really useful if the parent references are needed anyway, or stack space in particular is limited. For example, here is an iterative algorithm for in-order traversal:

visit(root) { midvisit() { print current.value next := current.right if next == null next := current.parent } prev := null current := root while current ≠ null if prev == current.parent next := current.left if next == null midvisit() else if prev == current.left midvisit() else next := current.parent prev := current current := next }

A similar scheme using a queue can effect level-order traversal, where we visit the nodes in each level of the tree from left to right. However, the space used is no longer proportional to the tree height but to the maximum tree level size, which can be as large as n/2. A more space-efficient approach for this type of traversal can be implemented using iterative deepening depth-first search.

visit(root) { q := empty queue add root to the end of q while q is not empty { node := get and remove first element of q print node.value if node.left ≠ null add node.left to the end of q if node.right ≠ null add node.right to the end of q } }

External links

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• The Adjacency List Model for Processing Trees with SQL
• Storing Hierarchical Data in a Database with traversal examples in PHP
• Managing Hierarchical Data in MySQL

Trees (structure)

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