Chapter 10
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Transcript of Chapter 10
Data Structures Using Java 2
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• Explore nonrecursive binary tree traversal
algorithms• Learn about AVL (height-balanced) trees
Data Structures Using Java 3
Binary Trees
• Definition: A binary tree, T, is either empty or such that:– 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 of T, respectively;
– LT and RT are binary trees
Data Structures Using Java 5
Binary Tree with One Node
The root node of the binary tree = A
LA = empty
RA = empty
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Binary Trees
Following class defines the node of a binary tree:
protected class BinaryTreeNode{ DataElement info; BinaryTreeNode llink; BinaryTreeNode rlink;}
Data Structures Using Java 10
Nodes
• For each node:– Data is stored in info
– The reference to the left child is stored in llink
– The reference to the right child is stored in rlink
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Binary Tree Definitions
• Leaf: node that has no left and right children• Parent: node with at least one child node• Level of a node: number of branches on the path
from root to node• Height of a binary tree: number of nodes no the
longest path from root to node
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Height of a Binary Tree
Recursive algorithm to find height of binary tree: (height(p) denotes height of binary tree with root p):
if(p is NULL)
height(p) = 0
else
height(p) = 1 + max(height(p.llink),height(p.rlink))
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Height of a Binary Tree
Method to implement above algorithm:
private int height(BinaryTreeNode p){ if(p == NULL) return 0; else return 1 + max(height(p.llink),
height(p.rlink));}
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Copy Tree
• Useful operation on binary trees is to make identical copy of binary tree
• Method copy useful in implementing copy constructor and method copyTree
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Method copyBinaryTreeNode copy(BinaryTreeNode otherTreeRoot){ BinaryTreeNode temp; if(otherTreeRoot == null) temp = null; else { temp = new BinaryTreeNode(); temp.info = otherTreeRoot.info.getCopy(); temp.llink = copy(otherTreeRoot.llink); temp.rlink = copy(otherTreeRoot.rlink); } return temp;}//end copy
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Method copyTree
public void copyTree(BinaryTree otherTree)
{
if(this != otherTree) //avoid self-copy
{
root = null;
if(otherTree.root != null) //otherTree is //nonempty
root = copy(otherTree.root);
}
}
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Binary Tree Traversal
• Must start with the root, then– Visit the node first
or
– Visit the subtrees first
• Three different traversals– Inorder
– Preorder
– Postorder
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Traversals
• Inorder – Traverse the left subtree– Visit the node– Traverse the right subtree
• Preorder– Visit the node– Traverse the left subtree– Traverse the right subtree
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Traversals
• Postorder– Traverse the left subtree
– Traverse the right subtree
– Visit the node
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Binary Tree: Inorder Traversal
private void inorder(BinaryTreeNode p){ if(p != NULL) { inorder(p.llink); System.out.println(p.info + “ “); inorder(p.rlink); }}
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Binary Tree: Preorder Traversal
private void preorder(BinaryTreeNode p){ if(p != NULL) {
System.out.println(p.info + “ “); preorder(p.llink);
preorder(p.rlink); }}
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Binary Tree: Postorder Traversal
private void postorder(BinaryTreeNode p){ if(p != NULL) { postorder(p.llink); postorder(p.rlink); System.out.println(p.info + “ “); }}
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Implementing Binary Trees: class BinaryTree methods
• isEmpty
• inorderTraversal
• preorderTraversal
• postorderTraversal
• treeHeight
• treeNodeCount
• treeLeavesCount
• destroyTree
• copyTree
• Copy
• Inorder
• Preorder
• postorder
• Height
• Max
• nodeCount
• leavesCount
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Binary Search Trees
• Data in each node– Larger than the data in its left child– Smaller than the data in its right child
• A binary search 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 of T, respectively– Key in root node larger than every key in left subtree
and smaller than every key in right subtree– LT and RT are binary search trees
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Operations Performed on Binary Search Trees
• Determine whether the binary search tree is empty• Search the binary search tree for a particular item• Insert an item in the binary search tree• Delete an item from the binary search tree
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Operations Performed on Binary Search Trees
• Find the height of the binary search tree• Find the number of nodes in the binary search tree• Find the number of leaves in the binary search tree• Traverse the binary search tree• Copy the binary search tree
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Binary Search Tree: Analysis
• Theorem: Let T be a binary search tree with n nodes, where n > 0.The average number of nodes visited in a search of T is approximately 1.39log2n
• Number of comparisons required to determine whether x is in T is one more than the number of comparisons required to insert x in T
• Number of comparisons required to insert x in T same as the number of comparisons made in unsuccessful search, reflecting that x is not in T
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Binary Search Tree: Analysis
It follows that:
It is also known that:
Solving Equations (10-1) and (10-2)
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Nonrecursive Inorder Traversal: General Algorithm
1. current = root; //start traversing the binary tree at // the root node2. while(current is not NULL or stack is nonempty) if(current is not NULL) { push current onto stack; current = current.llink; } else { pop stack into current; visit current; //visit the node current = current.rlink; //move to the right child
}
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Nonrecursive Preorder Traversal General Algorithm
1. current = root; //start the traversal at the root node2. 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 //the right subtree }
Data Structures Using Java 36
Nonrecursive Postorder Traversal
1. current = root; //start traversal at root node
2. v = 0;
3. if(current is NULL)
the binary tree is empty
4. if(current is not NULL)
a. push current into stack;
b. push 1 onto stack;
c. current = current.llink;
d. while(stack is not empty)
if(current is not NULL and v is 0){
push current and 1 onto stack;
current = current.llink;
}
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Nonrecursive Postorder Traversal (Continued)
else
{
pop stack into current and v;
if(v == 1)
{
push current and 2 onto stack;
current = current.rlink;
v = 0;
}
else
visit current;
}
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AVL (Height-Balanced Trees)
• A perfectly balanced binary tree is a binary tree such that:– The height of the left and right subtrees of the root are
equal
– The left and right subtrees of the root are perfectly balanced binary trees
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AVL (Height-Balanced Trees)
• An AVL tree (or height-balanced tree) is a binary search tree such that:– The height of the left and right subtrees of the root
differ by at most 1
– The left and right subtrees of the root are AVL trees
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AVL Tree Rotations
• Reconstruction procedure: rotating tree
• left rotation and right rotation
• Suppose that the rotation occurs at node x
• Left rotation: certain nodes from the right subtree of x move to its left subtree; the root of the right subtree of x becomes the new root of the reconstructed subtree
• Right rotation at x: certain nodes from the left subtree of x move to its right subtree; the root of the left subtree of x becomes the new root of the reconstructed subtree
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Deletion From AVL Trees
• Case 1: the node to be deleted is a leaf• Case 2: the node to be deleted has no right child,
that is, its right subtree is empty• Case 3: the node to be deleted has no left child,
that is, its left subtree is empty• Case 4: the node to be deleted has a left child and
a right child
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Analysis: AVL Trees
Consider all the possible AVL trees of height h. Let Th be an AVL tree of height h such that Th has the fewest number of nodes. Let Thl denote the left subtree of Th and Thr denote the right subtree of Th. Then:
where | Th | denotes the number of nodes in Th.
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Analysis: AVL Trees
Suppose that Thl is of height h – 1 and Thr is of height h – 2. Thl is an AVL tree of height h – 1 such that Thl has the fewest number of nodes among all AVL trees of height h – 1. Thr is an AVL tree of height h – 2 that has the fewest number of nodes among all AVL trees of height h – 2. Thl is of the form Th -1 and Thr is of the form Th -2. Hence:
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Analysis: AVL Trees
Let Fh+2 = |Th | + 1. Then:
Called a Fibonacci sequence; solution to Fh is given by:
Hence
From this it can be concluded that
Data Structures Using Java 59
Programming Example: Video Store (Revisited)
• In Chapter 4,we designed a program to help a video store automate its video rental process.
• That program used an (unordered) linked list to keep track of the video inventory in the store.
• Because the search algorithm on a linked list is sequential and the list is fairly large, the search could be time consuming.
• If the binary tree is nicely constructed (that is, it is not linear), then the search algorithm can be improved considerably.
• In general, item insertion and deletion in a binary search tree is faster than in a linked list.
• We will redesign the video store program so that the video inventory can be maintained in a binary tree.