data structures( 数据结构 ) course 8: search trees. 2 西南财经大学天府学院 chapter 8...
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Data Structures(Data Structures( 数据结数据结构构 ))
Course 8: Search TreesCourse 8: Search Trees
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Chapter 8 search treesChapter 8 search trees
Binary search trees and AVL trees 8-1 Binary search treesProblem: Store ordered data in an array structure :
efficient search algorithm, inefficient insertion and deletion
Linked list: efficient insertion and deletion ,inefficient search
Binary search trees can satisfy both of themDefinition: the properties of a binary search tree1. All items in the left subtree are less than the root2. All items in the right subtree are greater than or
equal to the root3. Each subtree is itself a binary search tree.
(a)
(d)(c)
(b)
1717
6 19
17
196
143
17
6
3
17
19
22
(e)
Figure 8-2 binary search trees
(a)
(d)(c)
(b)
17
6 11
17
196
223
17
22 19
17
196
1511
Figure 8-3 invalid binary search trees
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binary search tree traversals Algorithms are identical to the ones in chapter 7
Operations on binary search Operations on binary search treestrees
binary search tree search algorithmsFind smallest nodeFind largest nodeFind requested node
Follow the left branches until we get to a leafAlgorithm findsmallestBST( val root <pointer>)To find the smallest node in a BSTPre root is a pointer to a nonempty BST or subtreeReturn address of smallest node1 if (root->left null) 1 return(root)2 endif3 return findsmallestBST (root->left)end findsmallestBST
Follow the right branches to the last node in the tree Algorithm findlargestBST( val root <pointer>)To find the largest node in a BSTPre root is a pointer to a nonempty BST or subtreeReturn address of largest node1 if (root->right null) 1 return(root)2 endif3 return findlargestBST (root->right)end findlargestBST
Assume: To find T, Comparing T with root, T<root go left T>root go rightAlgorithm searchBST(val root <pointer>, val argument <key>)Search a binary search tree for a given valuePre root is the root to a binary tree or subtree argument is the key value requested Return the node address if the value is foundNull if the node is not in the tree1 if (root is null) 1 return null2 end if3 if (argument < root->key ) 1 return searchBST (root->left, argument)4 else if (argument > root->key ) 1 return searchBST (root->right, argument)5 else 1 return root6 end if end searchBST
23
4418
2012
Figure 8-4 a binary search trees
5235
Preorder traversal: 23 18 12 20 44 35 52
Postorder traversal: 12 20 18 35 52 44 23
Inorder traversal : 12 18 20 23 35 44 52
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Insert nodeInsert node
Steps: 1. Follow the branches to an empty subtree
2. Insert the new node All inserts take place at a leaf or a leaflike node that has only one null branch
23
4418
2012 5235
After inserting 19
23
4418
2012 5235
19
After inserting 38
23
4418
2012 5235
19 38
After inserting 23
23
4418
2012 5235
19 3823
Note: node with equal values are found in the right subtree
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Iterative insert (algorithm)Iterative insert (algorithm)
1 if (root is null) 1 root = new2 else 1 pwalk = root 2 loop ( pwalk not null ) 1 parent = pwalk 2 if ( new->key < pwalk->key ) 1 pwalk = pwalk->left 3 else 1 pwalk = pwalk->right 4 end if 3 end loopLocation for new node found 4 if (new->key < parent->key ) 1 parent->left = new 5 else 1 parent->right = new 6 end if3 end if4 returnend insertBST
Algorithm insertBST ( ref root <pointer>, val new <pointer> )Insert node containing new data into BST using iterationPre root is address of first node in a BST new is address of node containing data to be insertedPost new node inserted into the tree
Algorithm 8-4 iterative binary search tree insert
23
18
12 20
19new parent
Final position when pwalk null
null
pwalk
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Recursive insert Recursive insert node(algorithm)node(algorithm)
Algorithm addBST ( ref root <pointer>, val new <pointer> )Insert node containing new data into BST using recursionPre root is address of current node in a BST new is address of node containing data to be insertedPost new node inserted into the tree1 if (root is null) 1 root = new2 else locate null subtree for insertion 1 if ( new->key < root->key ) 1 addBST (root->left, new ) 2 else 1 addBST (root->right, new ) 3 end if 3 end if4 returnend insertBST
algorithm 8-5 add node to BST recursively
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Delete nodeDelete node
1.locate it2.delete: there are 4 possible cases
The node has no children: set the delete node’s parent to null, recycle its memory , and returnHas only a right subtree: attach the right subtree to the delete node’s parent and recycle its memory.Has only a left subtree: attach the left subtree to the delete node’s parent and recycle its memory.Has two subtree: to delete a node from the middle of a tree, to keep the balance, we must find a node,…, two ways:
Find the largest node in the deleted node’s left subtree, move its data to replace the deleted node’s dataFind the smallest node in the deleted node’s right subtree, move its data to replace the deleted node’s data
Regardless of which logic we use, we will be moving data from a leaf or leaflike node that can be deleted.
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(c) Delete node (18) with no left child
23
18 44
before dltptr
12
after
52
23
12 44
52
(b) Delete node (44) with no left child
23
18 44
before dltptr
12
23
18 52
12
after
52
dltptr
(a) Delete node (44) that has no children
before 23
18 44
23
18
after
(d) Delete node (23) with two children
23
12 44
before dltptr
8
after
52
14
12 44
5214 8
tree
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1 if (root is null) 1 return false2 endif3 if(dltkey < root->data.key ) 1 return deleteBST (root->left,dltkey)4 elseif (dltkey > root->data.key ) 1 return deleteBST (root->right,dltkey)5 else //delete node found—test for leaf node 1 if (root->left null) 1 dltptr = root 2 root = root->right 3 recycle ( dltptr ) 4 return true 2 elseif ( root->right null ) 1 dltptr = root 2 root = root->left 3 recycle ( dltptr ) 4 return true 3 else // node to be deleted not a leaf (or leaflike), find largest node on left subtree 1 dltptr = root->left 2 loop ( dltptr->right not null ) 1 dltptr = dltptr->right // node found, move data and delete leaf node 3 root->data = dltptr->data 4 return deleteBST (root->left,dltptr->data.key) 4 end if6 endifend deleteBST
Algorithm deleteBST ( ref root <pointer>, val dltKey <key> ) Delete a node from a BST Pre root is pointer to tree containing data to be deleted dltkey is key of node to be deletedPost node deleted and memory recycled, if dltkey not found, root unchangedReturn true if node deleted, false if not found
Base case 1Node not find
Base case 2After node deleted
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SummarySummary
A binary search tree is a binary tree with the following properties:
All items in the left subtree are less than the root.All items in the right subtree are greater than or equal to the root.Each subtree is itself a binary search tree.The inorder traversal of the binary search tree produces an ordered list.In a binary search tree, the node with the larges value is the rightmost node. To find the largest node, we follow the right branches until we get to a null right pointer.To search for a value in a binary search tree ,we follow the left or right branch down the tree ,depending on the value of the new node, until we find a null subtree.
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SummarySummary
To delete a node from a subtree, we must consider four cases:
The node to be deleted has no children.The node to be deleted has only a left subtree.The node to be deleted has only a right subtree.The node to be deleted has two subtrees.
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ExercisesExercises
Create a binary search tree using the following data entered as sequence set :7,10,14,23,33,56,66,70,80Insert 44 and 50 into the tree created in above exerciseCreate a binary search tree using the following data entered as sequence set :70,60,80,50,65,75,85,45,55,90Delete node contain 60 and node contain 85 from the binary search tree above
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Team workTeam work
Create a binary search tree using the following data entered as sequence set :70,60,80,50,65,75,85,45,55,90, output the ordered listInsert 34 and 79 into the tree created in above exercise and Delete node contain 60 and node contain 85 from the binary search tree above