12.5. Binary Tree Traversals¶
12.5.1. Binary Tree Traversals¶
Often we wish to process a binary tree by "visiting" each of its nodes, each time performing a specific action such as printing the contents of the node. Any process for visiting all of the nodes in some order is called a traversal. Any traversal that lists every node in the tree exactly once is called an enumeration of the tree's nodes. Some applications do not require that the nodes be visited in any particular order as long as each node is visited precisely once. For other applications, nodes must be visited in an order that preserves some relationship.
12.5.1.1. Preorder Traversal¶
For example, we might wish to make sure that we visit any given node before we visit its children. This is called a preorder traversal.
Figure 12.5.1: A binary tree for traversal examples.
Example 12.5.1
The preorder enumeration for the tree of Figure 12.5.1 is A B D C E G F H I.
The first node printed is the root. Then all nodes of the left subtree are printed (in preorder) before any node of the right subtree.
12.5.1.2. Postorder Traversal¶
Alternatively, we might wish to visit each node only after we visit its children (and their subtrees). For example, this would be necessary if we wish to return all nodes in the tree to free store. We would like to delete the children of a node before deleting the node itself. But to do that requires that the children's children be deleted first, and so on. This is called a postorder traversal.
Example 12.5.2
The postorder enumeration for the tree of Figure 12.5.1 is D B G E H I F C A.
12.5.1.3. Inorder Traversal¶
An inorder traversal first visits the left child (including its entire subtree), then visits the node, and finally visits the right child (including its entire subtree). The binary search tree makes use of this traversal to print all nodes in ascending order of value.
Example 12.5.3
The inorder enumeration for the tree of Figure 12.5.1 is B D A G E C H F I.
12.5.1.4. Implementation¶
Now we will discuss some implementations for the traversals, but we
need to define a node ADT to work with.
Just as a linked list is composed of a collection of link objects, a
tree is composed of a collection of node objects.
Here is an ADT for binary tree nodes, called BinNode.
This class will be used by some of the binary tree structures
presented later.
Member functions are provided that set or return the element value,
return a pointer to the left child,
return a pointer to the right child,
or indicate whether the node is a leaf.
interface BinNode { // Binary tree node ADT
// Get and set the element value
public Object value();
public void setValue(Object v);
// return the children
public BinNode left();
public BinNode right();
// return TRUE if a leaf node, FALSE otherwise
public boolean isLeaf();
}
interface BinNode { // Binary tree node ADT
// Get and set the element value
Object value();
void setValue(Object v);
// return the children
BinNode left();
BinNode right();
// return TRUE if a leaf node, FALSE otherwise
boolean isLeaf();
}
interface BinNode<E> { // Binary tree node ADT
// Get and set the element value
public E value();
public void setValue(E v);
// return the children
public BinNode<E> left();
public BinNode<E> right();
// return TRUE if a leaf node, FALSE otherwise
public boolean isLeaf();
}
A traversal routine is naturally written as a recursive
function.
Its input parameter is a pointer to a node which we will call
rt because each node can be viewed as the root of a some
subtree.
The initial call to the traversal function passes in a pointer to the
root node of the tree.
The traversal function visits rt and its children (if any)
in the desired order.
For example, a preorder traversal specifies that rt be
visited before its children.
This can easily be implemented as follows.
static void preorder(BinNode rt) {
if (rt == null) return; // Empty subtree - do nothing
visit(rt); // Process root node
preorder(rt.left()); // Process all nodes in left
preorder(rt.right()); // Process all nodes in right
}
void preorder(BinNode rt) {
if (rt == null) return; // Empty subtree - do nothing
visit(rt); // Process root node
preorder(rt.left()); // Process all nodes in left
preorder(rt.right()); // Process all nodes in right
}
static <E> void preorder(BinNode<E> rt) {
if (rt == null) return; // Empty subtree - do nothing
visit(rt); // Process root node
preorder(rt.left()); // Process all nodes in left
preorder(rt.right()); // Process all nodes in right
}
Function preorder first checks that the tree is not
empty (if it is, then the traversal is done and preorder
simply returns).
Otherwise, preorder makes a call to visit,
which processes the root node (i.e., prints the value or performs
whatever computation as required by the application).
Function preorder is then called recursively on the left
subtree, which will visit all nodes in that subtree.
Finally, preorder is called on the right subtree,
visiting all nodes in the right subtree.
Postorder and inorder traversals are similar.
They simply change the order in which the node and its children are
visited, as appropriate.
