Solution for Programmming Exercise 9.4
This page contains a sample solution to one of the exercises from Introduction to Programming Using Java.
Exercise 9.4:
Subsection 9.4.1 explains how to use recursion to print out the items in a binary tree in various orders. That section also notes that a non-recursive subroutine can be used to print the items, provided that a stack or queue is used as an auxiliary data structure. Assuming that a queue is used, here is an algorithm for such a subroutine:
Add the root node to an empty queue
while the queue is not empty:
Get a node from the queue
Print the item in the node
if node.left is not null:
add it to the queue
if node.right is not null:
add it to the queue
Write a subroutine that implements this algorithm, and write a program to test the subroutine. Note that you will need a queue of TreeNodes, so you will need to write a class to represent such queues.
(Note that the order in which items are printed by this algorithm is different from all three of the orders considered in Subsection 9.4.1.)
There's really not a lot to think about here, since such a complete algorithm is given. However, we do have to assemble the pieces. I use the standard binary tree node from Section 9.4 (except that I changed the name of the tree node class to StrTreeNode). The algorithm needs a queue of tree nodes. To implement this, I copied the QueueOfInts class from Subsection 9.3.2 and changed the type of the items in the queue to StrTreeNode. I also changed the name to TreeQueue. I did this literally by copying the class from a Web browser window and pasting it into my source code file. With these classes in hand, the algorithm given in the exercise can be coded as:
/**
* Use a queue to print all the strings in the tree to which
* root points. (The nodes will be listed in "level order",
* that is: first the root, then children of the root, then
* grandchildren of the root, and so on.)
*/
static void levelOrderPrint(StrTreeNode root) {
if (root == null)
return; // There is nothing to print in an empty tree.
TreeQueue queue; // The queue.
queue = new TreeQueue();
queue.enqueue(root);
while ( queue.isEmpty() == false ) {
StrTreeNode node = queue.dequeue();
System.out.println( node.item );
if ( node.left != null )
queue.enqueue( node.left );
if ( node.right != null )
queue.enqueue( node.right );
}
} // end levelOrderPrint()
The name of this routine comes from the order in which it prints out the nodes of the tree. Think of the root of the tree as being on the top "level" of the tree, the children of the root on the second level, the children of the children of the root on the third level, and so on. Then the subroutine prints the items in level order. That is, all the nodes on one level are printed before any of the nodes on the next level. This is a consequence of the way the algorithm processes the items. As items from one level are removed from the queue and printed, their children (which are the nodes on the next level) are added to the back of the queue. Just after all the items from one level have been processed, the queue contains all the children of those items, ready to be processed next. Level-order tree traversals can't be done by recursion, and they are a standard application of queues.
To test my subroutine, I wanted a reasonably large tree whose structure I knew, so I could check whether the nodes are printed in the correct order. (Since you didn't know about level-order traversals until now, on the other hand, you should have been mainly concerned with checking that all the nodes in the tree are printed, period.) I decided to create the binary sort tree shown in Subsection 9.4.2. To do this, I copied the treeInsert()subroutine from that section and used it to add names to the tree in an order that would produce the tree I wanted. Finally, I called levelOrderPrint() to output the names from the tree. (It worked!)
By the way, you might notice that the levelOrderPrint() routine is very similar to the technique used in the grid-marking applet in the sample program DepthBreadth.java from Subsection 9.3.2. In fact they are just variations on the same idea. One difference is that in DepthBreadth.java, the squares of the grid had to be marked as "visited" as they were processed to avoid going into an infinite loop. The levelOrderPrint() subroutine doesn't have to do the same type of marking because it is working on a tree. One of the defining properties of a tree is that it cannot contain a loop of nodes. That is, it is not possible for a node to be its own descendent. This restriction guarantees that levelOrderPrint() will not go into an infinite loop. The same property guarantees that all of our recursive tree-processing methods will not suffer from infinite recursion when they are applied to a tree. You should note, however, that it is possible to connect tree nodes into data structures that contain loops and are therefore not trees at all. While these data structures are not trees, they might have other uses. Many of the subroutines we've looked will fail if applied to these loopy structures.
/**
* This program includes a non-recursive subroutine that prints the
* nodes of a binary tree, using a queue. The main program simply
* tests that routine. (The nodes are printed in what is called
* "level order".)
*
* This file defines the queue and tree classes as nested classes.
* Since they are general-purpose classes, it would really be better
* to put them in separate files.
*/
public class TreePrintNonRecursive {
//--------------------------------- NESTED CLASSES -----------------------
/**
* An object in this class is a node in a binary tree
* in which the nodes contain items of type String.
*/
static class StrTreeNode {
//
String item; // The item
StrTreeNode left; // Pointer to left subtree.
StrTreeNode right; // Pointer to right subtree.
StrTreeNode( String str ) {
// Constructor. Make a node to contain str.
item = str;
}
} // end class StrTreeNode
/**
* An object of this type represents a queue of StrTreeNodes,
* with the usual operations: dequeue, enqueue, isEmpty.
*/
static class TreeQueue {
/**
* An object of type Node holds one of the items
* in the linked list that represents the queue.
*/
private static class Node {
StrTreeNode item;
Node next;
}
private Node head = null; // Points to first Node in the queue.
// The queue is empty when head is null.
private Node tail = null; // Points to last Node in the queue.
/**
* Add N to the back of the queue.
*/
void enqueue( StrTreeNode tree ) {
Node newTail = new Node(); // A Node to hold the new item.
newTail.item = tree;
if (head == null) {
// The queue was empty. The new Node becomes
// the only node in the list. Since it is both
// the first and last node, both head and tail
// point to it.
head = newTail;
tail = newTail;
}
else {
// The new node becomes the new tail of the list.
// (The head of the list is unaffected.)
tail.next = newTail;
tail = newTail;
}
}
/**
* Remove and return the front item in the queue.
* Throws an IllegalStateException if the queue is empty.
*/
StrTreeNode dequeue() {
if ( head == null)
throw new IllegalStateException("Can't dequeue from an empty queue.");
StrTreeNode firstItem = head.item;
head = head.next; // The previous second item is now first.
if (head == null) {
// The queue has become empty. The Node that was
// deleted was the tail as well as the head of the
// list, so now there is no tail. (Actually, the
// class would work fine without this step.)
tail = null;
}
return firstItem;
}
/**
* Return true if the queue is empty, false if contains one
* or more items
*/
boolean isEmpty() {
return (head == null);
}
} // end class TreeQueue
//-------------------- END OF NESTED CLASSES ---------------------------
static StrTreeNode root; // A pointer to the root of the binary tree.
/**
* Use a queue to print all the strings in the tree to which
* root points. (The nodes will be listed in "level order",
* that is: first the root, then children of the root, then
* grandchildren of the root, and so on.)
*/
static void levelOrderPrint(StrTreeNode root) {
if (root == null)
return; // There is nothing to print in an empty tree.
TreeQueue queue; // The queue, which will only hold non-null nodes.
queue = new TreeQueue();
queue.enqueue(root);
while ( queue.isEmpty() == false ) {
StrTreeNode node = queue.dequeue();
System.out.println( node.item );
if ( node.left != null )
queue.enqueue( node.left );
if ( node.right != null )
queue.enqueue( node.right );
}
} // end levelOrderPrint()
/**
* Add the word to the binary sort tree to which the
* global variable "root" refers. I will use this
* routine only to create the sample tree on which
* I will test levelOrderPrint().
*/
static void treeInsert(String newItem) {
if ( root == null ) {
// The tree is empty. Set root to point to a new node
// containing the new item.
root = new StrTreeNode( newItem );
return;
}
StrTreeNode runner; // Runs down the tree to find a place for newItem.
runner = root; // Start at the root.
while (true) {
if ( newItem.compareTo(runner.item) < 0 ) {
// Since the new item is less than the item in runner,
// it belongs in the left subtree of runner. If there
// is an open space at runner.left, add a node there.
// Otherwise, advance runner down one level to the left.
if ( runner.left == null ) {
runner.left = new StrTreeNode( newItem );
return; // New item has been added to the tree.
}
else
runner = runner.left;
}
else {
// Since the new item is greater than or equal to the
// item in runner, it belongs in the right subtree of
// runner. If there is an open space at runner.right,
// add a new node there. Otherwise, advance runner
// down one level to the right.
if ( runner.right == null ) {
runner.right = new StrTreeNode( newItem );
return; // New item has been added to the tree.
}
else
runner = runner.right;
}
} // end while
} // end treeInsert()
/**
* Make a tree with a known form, then call levelOrderPrint()
* for that tree. (I want to check that all the items from
* the tree are printed, and I want to see the order in which
* they are printed. The expected order of output is
* judy bill mary alice fred tom dave jane joe. The
* tree that is built here is from an illustration in
* Section 11.4.)
*/
public static void main(String[] args) {
treeInsert("judy");
treeInsert("bill");
treeInsert("fred");
treeInsert("mary");
treeInsert("dave");
treeInsert("jane");
treeInsert("alice");
treeInsert("joe");
treeInsert("tom");
levelOrderPrint(root);
} // end main()
} // end class TreePrintNonRecursive