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Ceng-112 Data Structures I 2006 1
Chapter 8
SearchTrees
Ceng-112 Data Structures I 2006 2
Binary Search Trees
• The binary search is a very efficient search algorithm, without
insertion and deletion, when we store ordered data in the array
structure.
• When we use linked list, in that case we have to use sequential
search as an inefficient search algorithm.
• Binary search trees provide a suitable structure.
Ceng-112 Data Structures I 2006 3
Figure 8-1
Binary Search Trees Definition
1. All terms in the left subtree are less than the root.2. All terms in the right subtree are greater than or equal to the root.3. Each subtree is itself a binary search tree.
< K
Ceng-112 Data Structures I 2006 4
Figure 8-2
Binary Search Trees
Ceng-112 Data Structures I 2006 5
Figure 8-3
Binary Search Trees
Ceng-112 Data Structures I 2006 6
Figure 8-4
Binary Search Trees
Preorder traversal?Postorder traversal?Inorder traversal?
Ceng-112 Data Structures I 2006 7
Binary Search Trees Binary Search TreesFind the smallest value
algorithm findsmallestBST(val root <pointer>)
This algorithm finds the smallest node in a BST.
PRE root is a pointer to a non-empty BST.
Return address of smallest node
1. if (root->left null)
1. return (root)
2. return findsmallestBST(root->left)
end findsmallestBST
Ceng-112 Data Structures I 2006 8
Figure 8-5
Binary Search Trees
Find the smallest node in BST.
Ceng-112 Data Structures I 2006 9
Binary Search Trees Binary Search TreesFind the largest value
algorithm findlargestBST(val root <pointer>)
This algorithm finds the larges node in a BST.
PRE root is a pointer to a non-empty BST.
Return address of largest node
1. if (root->right null)
1. return (root)
2. return findlargestBST(root->right)
end findlargestBST
Ceng-112 Data Structures I 2006 10
Figure 8-6
Binary Search Tree Search
Finding a specific node in the tree!
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Figure 8-7
We are looking for node 20.
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Binary Search Trees Binary Search TreesFind the specific value
algorithm searchBST(val root <pointer>, val argument <key>)
Search a binary search tree for a given value.
PRE root is the root to a binary tree or subtree, argument is the key value requested.
Return the node address if the value is found, null if the node is not in the tree.
1 If (root is null)
1 return null
2 If (argument < root->key)
1 return searchBST(root->left, argument)
3 else If (argument > root->key)
1 return searchBST(root->right, argument)
4 else
1 return root
end searchBST
Ceng-112 Data Structures I 2006 13
Binary Search Tree – Insert Node
Ceng-112 Data Structures I 2006 14
Binary Search Tree – Insert Node
algorithm insertBST(ref root <pointer>, val new <pointer>)
Insert node containing new node into BST using iteration
PRE root is address of the first node in a BST, new is address
of node containing data to be inserted.
POST new node inserted into tree.
Ceng-112 Data Structures I 2006 15
Binary Search Tree – Insert Node1 If (root is null)
1 root = new2 else
1 pwalk = root2 loop (pwalk not null)
1 parent =pwalk2 If (new->key < pwalk->key)
1 pwalk=pwalk->left3 else pwalk=pwalk->right
Location for new node found3 If (new->key < parent->key)
1 parent->left=new4 else parent->right=new
3 returnend insertBST
Ceng-112 Data Structures I 2006 16
Binary Search Tree – Delete Node
Ceng-112 Data Structures I 2006 17
Binary Search Tree – Delete Node
Ceng-112 Data Structures I 2006 18
Binary Search Tree – Delete Node
algorithm deleteBST(ref root <pointer>, val dltkey <key>)
This algorithm deletes a node from BST.
Pre root is pointer to tree containing data to be deleted,
dltkey is key of node to be deleted.
Post node deleted & memory rcycled, if dltkey not found, root
unchanged.
Return true if node deleted, false if not found.
Ceng-112 Data Structures I 2006 19
Binary Search Tree – Delete Node
1 If (root null)1 return false
2 If (dltkey < root->key) return deleteBST(root->left, dltkey)3 else If (dltkey > root->key) return deleteBST(root->right, dltkey)4 else /*(Delete node found --- Test for leaf node)*/
1 If (root ->left null)1 dltprt=root, root =root->right, recycle(dltprt), return true
2 else If (root->right null)1 dltprt=root, root =root->left, recycle(dltprt), return true
3 else /*Node is not a leaf, find largest node on left subtree*/1 dltprt = root->left2 loop (dltprt->right not null)
1 dltprt=dltprt->right3 root->data =dltprt->data4 return deleteBST(root->left, dltprt->data.key)
end deleteBST
Ceng-112 Data Structures I 2006 20
HW-8
1. Create a BST with positive integer numbers which are taken from the screen.
2. Write the BST delete function which establishes to delete desired node from the BST.
3. Write the BST list function which lists the nodes in the BST with inorder traversal.
4. Collect all above functions under a user menu.
Load your HW-8 to FTP site until 14 May. 07 at 09:00 am.
Ceng-112 Data Structures I 2006 21
AVL Trees
In 1962, two Russian mathematicians, G. M. Adelson-Velskil and E. M. Landis created balanced binary tree structure that is named after them – the AVL trees.
|HL-HR| < = 1Height balanced trees.
O(n) O(log2n)
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Ceng-112 Data Structures I 2006 23
Figure 8-14 (a and b)
When we insert a node into a tree or delete a node from a tree, the resulting tree may be unbalanced and we must rebalance it.
AVL Trees – Balancing
Ceng-112 Data Structures I 2006 24
All unbalanced trees fall into one of these four cases:1. Left to left2. Right to right3. Right of left4. Left of right
Ceng-112 Data Structures I 2006 25
Figure 8-15
LEFT-OF-LEFTWe must balance the left-height
tree by rotating the out-of-balancenode to the right.
Ceng-112 Data Structures I 2006 26
Figure 8-16
RIGHT-OF-RIGHTWe must balance the right-heighttree by rotating the out-of-balance
node to the left.
Ceng-112 Data Structures I 2006 27
Figure 8-17
RIGHT-OF-LEFT1. Rotate left2. Rotate right
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Figure 8-18
LEFT-OF-RIGHT1. Rotate right2. Rotate left
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AVL Node Structure
Node
key <key type>
data <data type>
leftsubtree <pointer to Node>
rightsubtree <pointer to Node>
bal <LH, EH, RH>
End Node
Ceng-112 Data Structures I 2006 30
AVL Rotate Algorithm
algorithm rotateRight( ref root <tree pointer>)
This algorithm exchanges pointers to rotate the tree right.
PRE root points to tree to be rotated.
POST Node rotated and root updated.
1 tempPtr = root->left
2 root->left = tempPtr->right
3 tempPtr->right = root
4 root = tempPtr
5 return
end rotateRight
Ceng-112 Data Structures I 2006 31
AVL Insert
• The search and retrieval algorithms are the same as for any
binary tree.
• Inorder travelsal is used because AVL trees are search trees.
• As a binary search tree, we have to find suitable leaf node on left
or right subtree, then we connect new node to this parent node
and begin back out of tree.
• As we back out of tree we check the balance of each node. If we
find unbalanced node we balance it and continue up the tree.
• Not all inserts create an out of balance condition.
Ceng-112 Data Structures I 2006 32
AVL Insert Algorithm
algoritm AVLInsert (ref root <tree pointer>,
ref newPtr <tree pointer>,
ref taller <boolean>)
Using recursion, insert a node into AVL tree.
PRE root is a pointer to first node in AVL tree/subtree
newPtr is a pointer to new node to be inserted.
POST taller is a boolean: true indicating the subtree height has
increased, false indicating same height.
Ceng-112 Data Structures I 2006 33
AVL Insert Algorithm1 if (root null) // Insert at root //
1 taller = true 2 root= newPtr2 else
1 if (newPtr->key < root->key) //left subtree //1 AVLInsert(root->left, newPtr, taller)2 if (taller) // insertion is completed and height is changed //
1 if (root left-height)1 leftBalance(root, taller)
2 else if (root right-height)1 taller =false
3 adjust balance factor2 else if (newPtr->key > root->key) //right subtree //
1 AVLInsert(root->right, newPtr, taller)2 if (taller) // insertion is completed and height is changed //
1 if (root left-height)1 taller =false
2 else if (root right-height)1 rihtBalance(root, taller)
3 adjust balance factor3 else
1 error (“Dupe Data”)2 recycle(newPtr)3 taller = false
3 returnend AVLInsert
Ceng-112 Data Structures I 2006 34
AVL Left Balance Algorithmalgorithm leftBalance(ref root <tree pointer>, ref taller <boolean>)This algorithm is entered when the root is left heavy (the left subtree is higher then
the right subtree)PRE root is a pointer to the root of the (sub)tree, taller is true.POST root and taller has been updated.1 leftTree = root->left2 if (leftTree left-heigh) //Case 1: Left of left, single rotation required. //
1 rotateRight (root)2 adjust balance factors3 taller false
3 else // Case 2: Right of left. Double rotation required. //1 rightTree = leftTree->right2 adjust balance factors3 rotateLeft (leftTree)4 rotateRight (root)5 taller = false
4 returnend leftBalance
Ceng-112 Data Structures I 2006 35
Figure 8-22
AVL Delete Balancing
Ceng-112 Data Structures I 2006 36
Figure 8-25
Which of the trees in the following figure is a valid binary search treeand which one is not?
Excercise
Ceng-112 Data Structures I 2006 37
Figure 8-26
Travers the binary search tree using a inorder traversal.
Excercise
Ceng-112 Data Structures I 2006 38
Figure 8-27
The binary search tree was created starting with a null tree and enteringdata from the keyboard. In what sequence were the data entered? If there is more than one possible sequence, identify the alternatives.
Excercise
Ceng-112 Data Structures I 2006 39
Figure 8-28
The binary search tree was created starting with a null tree and enteringdata from the keyboard. In what sequence were the data entered? If there is more than one possible sequence, identify the alternatives.
Excercise
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Figure 8-29
Insert 44, 66 and 77 in the binary search tree.
Excercise
Ceng-112 Data Structures I 2006 41
Figure 8-30
Delete the node 60 and then 85 from the tree.
Excercise
Ceng-112 Data Structures I 2006 42
Figure 8-31
Balance the tree.
Excercise
Ceng-112 Data Structures I 2006 43
Figure 8-32
Balance the tree.
Excercise
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Figure 8-33
Add 49 to the AVL tree. The result must be an AVL tree. Show the balance factors in the resulting tree.
Excercise
Ceng-112 Data Structures I 2006 45
Figure 8-33
Add 68 to the AVL tree. The result must be an AVL tree. Show the balance factors in the resulting tree.
Excercise
Ceng-112 Data Structures I 2006 46
HW-9
1. Create an AVL tree with positive integer numbers which are taken from the screen.
2. Write the AVL delete function which establishes to delete desired node from the AVL.
3. Write the AVL list function which lists the nodes in the AVL with inorder traversal.
4. Collect all above functions under a user menu.
Load your HW-9 to FTP site until 14 May. 07 at 09:00 am.