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Chapter 19: Binary Trees
Java Programming:
Program Design Including Data Structures
Chapter Objectives




Learn about binary trees
Explore various binary tree traversal algorithms
Learn how to organize data in a binary search tree
Discover how to insert and delete items in a binary
search tree
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Binary Trees
 A binary tree, T, is either empty or
 T has a special node called the root node
 T has two sets of nodes, LT and RT, called the left
subtree and right subtree
 LT and RT are binary trees
 A binary tree can be shown pictorially
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Binary Trees (continued)
Figure 19-1 Binary tree
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Binary Trees (continued)
Figure 19-6 Various binary trees with three nodes
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Binary Trees (continued)
 You can write a class that represents each node in a
binary tree
 Called BinaryTreeNode
 Instance variables of the class BinaryTreeNode
 info: stores the information part of the node
 lLink: points to the root node of the left subtree
 rLink: points to the root node of the right subtree
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Binary Trees (continued)
Figure 19-7 UML class diagram of the class BinaryTreeNode and the
outer-inner class relationship
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Binary Trees (continued)
Figure 19-8 Binary tree
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Binary Trees (continued)
 A leaf is a node in a tree with no children
 Let U and V be two nodes in a binary tree
 U is called the parent of V if there is a branch from U
to V
 A path from a node X to a node Y is a sequence of
nodes X0, X1, ..., Xn such that:
 X = X0,Xn = Y
 Xi-1 is the parent of Xi for all i = 1, 2, ..., n
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Binary Trees (continued)
 Length of a path
 The number of branches on that path
 Level of a node
 The number of branches on the path from the root to
the node
 Height of a binary tree
 The number of nodes on the longest path from the
root to a leaf
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Binary Trees (continued)
 Method height
private int height(BinaryTreeNode<T> p)
{
if (p == null)
return 0;
else
return 1 + Math.max(height(p.lLink), height(p.rLink));
}
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Copy Tree
 Method copyTree
private BinaryTreeNode<T> copyTree
(BinaryTreeNode<T> otherTreeRoot)
{
BinaryTreeNode<T> temp;
if (otherTreeRoot == null)
temp = null;
else
{
temp = (BinaryTreeNode<T>) otherTreeRoot.clone();
temp.lLink = copyTree(otherTreeRoot.lLink);
temp.rLink = copyTree(otherTreeRoot.rLink);
}
return temp;
}//end copyTree
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Binary Tree Traversal
 Item insertion, deletion, and lookup operations
require that the binary tree be traversed
 Commonly used traversals
 Inorder traversal
 Preorder traversal
 Postorder traversal
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Inorder Traversal
 Binary tree is traversed as follows:
 Traverse left subtree
 Visit node
 Traverse right subtree
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Inorder Traversal (continued)
 Method inOrder
private void inorder(BinaryTreeNode<T> p)
{
if (p != null)
{
inorder(p.lLink);
System.out.print(p + “ “);
inorder(p.rLink);
}
}
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Preorder Traversal
 Binary tree is traversed as follows:
 Visit node
 Traverse left subtree
 Traverse right subtree
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Preorder Traversal (continued)
 Method preOrder
private void preorder(BinaryTreeNode<T> p)
{
if (p != null)
{
System.out.print(p + “ “);
preorder(p.lLink);
preorder(p.rLink);
}
}
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Postorder Traversal
 Binary tree is traversed as follows:
 Traverse left subtree
 Traverse right subtree
 Visit node
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Postorder Traversal (continued)
 Method postOrder
private void postorder(BinaryTreeNode<T> p)
{
if (p != null)
{
postorder(p.lLink);
postorder(p.rLink);
System.out.print(p + “ “);
}
}
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Implementing Binary Trees
Figure 19-11 UML class diagram of the interface BinaryTreeADT
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Implementing Binary Trees
(continued)
Figure 19-12 UML class diagram of the class BinaryTree
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Implementing Binary Trees
(continued)
 Method clone
public Object clone()
{
BinaryTree<T> copy = null;
try
{
copy = (BinaryTree<T>) super.clone();
}
catch (CloneNotSupportedException e)
{
return null;
}
if (root != null)
copy.root = copyTree(root);
return copy;
}
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Binary Search Trees
 To search for an item in a normal binary tree, you
must traverse entire tree until item is found
 Search process will be very slow
 Similar to searching in an arbitrary linked list
 Binary search tree
 Data in each node is
 Larger than the data in its left child
 Smaller than the data in its right child
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Binary Search Trees (continued)
Figure 19-14 Binary search tree
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Binary Search Trees (continued)
Figure 19-15 UML class diagram of the class BinarySearchTree
and the inheritance hierarchy
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Search
 Searches tree for a given item
 General steps
 Compare item with info in root node
 If they are the same, stop the search
 If item is smaller than info in root node
 Follow link to left subtree
 Otherwise
 Follow link to right subtree
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Insert
 Inserts a new item into a binary search tree
 General steps
 Search tree and find the place where new item is to be
inserted
 Search algorithm is similar to method search
 Insert new item
 Duplicate items are not allowed
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Delete
 Deletes item from a binary search tree
 After deleting items, resulting tree must be a binary
search tree
 General steps
 Search the tree for the item to be deleted
 Searching algorithm is similar to method search
 Delete item
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Delete (continued)
 Delete operation has four cases




Case 1: Node to be deleted is a leaf
Case 2: Node to be deleted has no left subtree
Case 3: Node to be deleted has no right subtree
Case 4: Node to be deleted has nonempty left and
right subtrees  search left subtree of the node to be
deleted to find its immediate predecessor
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Binary Search Tree: Analysis
 Performance depends on shape of the tree
 If tree shape is linear, performance is the same as for
a linked list
 Average number of nodes visited
1.39log2n = O(log2n)
 Average number of key comparisons
2.77log2n = O(log2n)
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Nonrecursive Binary Tree
Traversal Algorithms
 Traversal algorithms
 Inorder
 Preorder
 Postorder
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Nonrecursive Inorder Traversal
 Method nonRecursiveInTraversal
public void nonRecursiveInTraversal()
{
LinkedStackClass<BinaryTreeNode<T> > stack
= new LinkedStackClass<BinaryTreeNode<T> >();
BinaryTreeNode<T> current;
current = root;
while ((current != null) || (!stack.isEmptyStack()))
if (current != null)
{
stack.push(current);
current = current.lLink;
}
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Nonrecursive Inorder Traversal
(continued)
else
{
current = (BinaryTreeNode<T>) stack.peek();
stack.pop();
System.out.print(current.info + “ “);
current = current.rLink;
}
System.out.println();
}
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Nonrecursive Preorder Traversal
 General algorithm
create stack
current = root;
while (current is not null or stack is nonempty)
if (current is not null)
{
visit current;
push current onto stack;
current = current.lLink;
}
else
{
pop stack into current;
current = current.rLink; //prepare to visit right subtree
}
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Nonrecursive Postorder Traversal
 General algorithm





Mark left subtree of node as visited
Visit left subtree and return to node
Mark right subtree of node as visited
Visit right subtree and return to node
Visit node
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An Iterator to a Binary Tree
 You can create an iterator for a binary tree
 Iterator can traverse tree using
 Inorder traversal
 Preorder traversal
 Postorder traversal
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AVL (Height-Balanced) Trees
 Search performance depends on shape of the tree
 You want the tree to be balanced
 AVL (height-balanced) tree
 Resulting binary search tree is nearly balanced
 Perfectly balanced binary search tree
 Heights of the left and right subtrees are equal
 Left and right subtrees are perfectly balanced binary
trees
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AVL (Height-Balanced)
Trees(continued)
Figure 19-24 Perfectly balanced binary tree
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AVL (Height-Balanced) Trees
(continued)
 AVL tree: A binary search tree where
 Heights of left and right subtrees differs by at most 1
 Left and right subtrees are AVL trees
 Balance factor
 Difference between height of right subtree and height
of left subtree
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AVL (Height-Balanced) Trees
(continued)
Figure 19-25 Perfectly balanced binary tree
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Insertion into AVL Trees
 General steps
 Search AVL tree to find insertion point
 Insert new item
 Duplicate items are not allowed
 Rebalance tree (if needed)
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Insertion into AVL Trees
(continued)
Figure 19-27 AVL tree before inserting 90
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Insertion into AVL Trees
(continued)
Figure 19-28 Binary search tree of Figure 19-27 after inserting 90; nodes other
than 90 show their balance factors before insertion
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Insertion into AVL Trees
(continued)
Figure 19-29 AVL tree of Figure 19-27 after inserting 90 and adjusting the
balance factors
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AVL Tree Rotations
 Reconstruction procedure
 Types of rotations
 Left rotation
 Nodes from right subtree move to left subtree
 Root of right subtree becomes root of reconstructed
subtree
 Right rotation
 Nodes from left subtree move to right subtree
 Root of left subtree becomes root of reconstructed
subtree
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AVL Tree Rotations (continued)
Figure 19-39 Left rotation at a
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AVL Tree Rotations (continued)
Figure 19-39 Right rotation at b
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AVL Tree Rotations (continued)
 Method rotateToLeft
private AVLNode<T> rotateToLeft(AVLNode<T> root)
{
AVLNode<T> p; //reference variable to the root of the right subtree of root
if (root == null)
System.err.println(“Error in the tree.”);
else
if (root.rLink == null)
System.err.println(“Error in the tree: “
+ “No right subtree to rotate.”);
else
{
p = root.rLink;
root.rLink = p.lLink; //the left subtree of p becomes the right
//subtree of root
p.lLink = root;
root = p;
//make p the new root node
}
return root;
}//end rotateToLeft
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AVL Tree Rotations (continued)
 Method rotateToRight
private AVLNode<T> rotateToRight(AVLNode<T> root)
{
AVLNode<T> p; //reference variable to the root of the
//left subtree of root
if (root == null)
System.err.println(“Error in the tree.”);
else
if (root.lLink == null)
System.err.println(“Error in the tree: “
+ “No left subtree to rotate.”);
else
{
p = root.lLink;
root.lLink = p.rLink; //the right subtree of p
//becomes the left subtree of root
p.rLink = root;
root = p;
//make p the new root node
}
return root;
}//end rotateToRight
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AVL Tree Rotations (continued)
Figure 19-44 AVL tree after inserting 40
Figure 19-45 AVL tree after inserting 30
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AVL Tree Rotations (continued)
Figure 19-46 AVL tree after inserting 20
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AVL Tree Rotations (continued)
Figure 19-46 AVL tree after inserting 25
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Deletion from AVL Trees
 General steps
 Find node to be deleted
 Delete node
 Four cases arise:




Node to be deleted is a leaf
Node to be deleted has no right child
Node to be deleted has no left child
Node to be deleted has both children
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Analysis: AVL Trees
 Height of AVL tree with n nodes (worst case)
1.44log2n = O(log2n)
 Time to manipulate an AVL tree in the worst case is
no more than 44% of optimum time
 Average search time of an AVL tree is about 4%
more than the optimum
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Programming Example: Video Store
(Revisited)
 Program in Chapter 16 used a linked list to keep
track of video inventory
 Search could be time consuming
 Modify program to use a binary tree instead
 Insertion and deletion in a binary search tree is faster
than in a linked list
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Chapter Summary
 Binary trees
 Every node has only two children
 Left subtree
 Right subtree
 Binary tree traversal
 Inorder
 Preorder
 Postorder
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Chapter Summary (continued)
 Binary search trees
 Each node is greater than elements in its left subtree
and less than elements in its right subtree
 AVL (height-balanced) trees
 Binary search tree
 Heights of left and right subtrees differs by at most 1
 Left and right subtrees of the root node are AVL trees
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