Terminology. The ADT Binary Tree. The ADT Binary Search Tree
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1 Terminology The ADT Binary Tree The ADT Binary Search Tree 1
2 Terminology 3 A general tree A general tree T is a set of one or more nodes such that T is partitioned into disjoint subsets: o A single node r, the root o Sets that are general trees, called subtrees of r President Marketing Manufacturing Personnel Director Media Relations Director Sales 2
3 A general tree Parent of node n o The node directly above node n in the tree Child of node n o A node directly below node n in the tree Root o The only node in the tree with no parent Leaf o A node with no children A general tree Siblings o Nodes with a common parent Ancestor of node n o A node on the path from the root to n Descendant of node n o A node on a path from n to a leaf Subtree of node n o A tree that consists of a child (if any) of n and the child s descendants 3
4 A general tree Height o The number of nodes on the longest path from the root to a leaf Left (right) child of node n o A node directly below and to the left (right) of node n in a binary tree Left (right) subtree of node n o In a binary tree, the left (right) child (if any) of node n plus its descendants A general tree An n-ary tree o A generalization of tree whose nodes each can have no more than n children 4
5 A binary tree A binary tree is a set T of nodes such that either T is empty, or T is partitioned into three disjoint subsets: o A single node r, the root o Two possibly empty sets that are binary trees, called left and right subtrees of r A tree with each node has no more than two children binary tree Binary trees with the same nodes but different heights 5
6 A binary search tree A binary tree that has the following properties : for each node n o n s value is greater than all values in its left subtree T L o n s value is less than all values in its right subtree T R o Both T L and T R are binary search trees Full, complete, and balanced binary trees Full binary trees A full binary tree of height h, all nodes that are at level less than h have two children each 6
7 Full, complete, and balanced binary trees Complete binary trees A complete binary tree is a binary tree that is full down to level h-1, with level h filled in from left to right Full, complete, and balanced binary trees Balanced binary trees o A binary tree is height balanced or simple balance if the height of any node s right subtree differs from the height of the node s left subtree by no more than 1. Full binary trees are complete Complete binary trees are balanced 7
8 Full, complete, and balanced binary trees The ADT Binary Tree 16 8
9 The operations available for a particular ADT binary tree depend on the type of binary tree being implemented Basic operations of the ADT binary tree createbinarytree() createbinarytree(rootitem) makeempty() isempty() getrootitem() throws TreeException General operations of the ADT binary tree o createbinarytree (rootitem, lefttree, righttree) o setrootitem(newitem) o attachleft(newitem) throws TreeException o attachright(newitem) throws TreeException o attachleftsubtree(lefttree) throws TreeException o attachrightsubtree(righttree) throws TreeException o detachleftsubtree() throws TreeException o detachrightsubtree() throws TreeException 9
10 Traversals of a Binary Tree A traversal algorithm for a binary tree visits each node in the tree Recursive traversal algorithms o Preorder traversal o Inorder traversal o Postorder traversal Traversal is O(n) Traversals of a Binary Tree
11 Traversals of a Binary Tree Possible Representations of a Binary Tree An array-based representation o A Java class is used to define a node in the tree o A binary tree is represented by using an array of tree nodes o Each tree node contains a data portion and two indexes (one for each of the node s children) o Requires the creation of a free list which keeps track of available nodes 11
12 Possible Representations of a Binary Tree An array-based representation An array-based representation of a complete tree If the binary tree is complete and remains complete o A memory-efficient array-based implementation can be used 12
13 An reference-based representation of a complete tree A reference-based representation o Java references can be used to link the nodes in the tree The ADT Binary Search Tree 26 13
14 A deficiency of the ADT binary tree which is corrected by the ADT binary search tree Searching for a particular item Each node n in a binary search tree satisfies the following properties n s value is greater than all values in its left subtree T L n s value is less than all values in its right subtree T R Both T L and T R are binary search trees Record A group of related items, called fields, that are not necessarily of the same data type Field A data element within a record A data item in a binary search tree has a specially designated search key A search key is the part of a record that identifies it within a collection of records 14
15 Operations of the ADT binary search tree Insert a new item into a binary search tree Delete the item with a given search key from a binary search tree Retrieve the item with a given search key from a binary search tree Traverse the items in a binary search tree in preorder, inorder, or postorder Algorithms for the Operations of the ADT Binary Search Tree Since the binary search tree is recursive in nature, it is natural to formulate recursive algorithms for its operations A search algorithm o search(bst, searchkey) o Searches the binary search tree bst for the item whose search key is searchkey 15
16 Algorithms for the Operations of the ADT Binary Search Tree The shape of a binary tree affects search s efficiency Tom Alan Bob Janet Wendy Ellen Bob Karen Janet Alan Ellen Karen Tom Wendy Algorithms for the Operations of the ADT Binary Search Tree Insertion o insertitem(treenode, newitem) o Inserts newitem into the binary search tree of which treenode is the root 16
17 Algorithms for the Operations of the ADT Binary Search Tree Insertion o insertitem(treenode, newitem) o Inserts newitem into the binary search tree of which treenode is the root Algorithms for the Operations of the ADT Binary Search Tree Deletion o Steps for deletion o Use the search algorithm to locate the item with the specified key o If the item is found, remove the item from the tree o Three possible cases for node N containing the item to be deleted o N is a leaf o N has only one child o N has two children 17
18 Algorithms for the Operations of the ADT Binary Search Tree Deletion o Strategies for deleting node N o If N is a leaf o Set the reference in N s parent to null o If N has only one child o Let N s parent adopt N s child o If N has two children o Locate another node M that is easier to remove from the tree than the node N o Copy the item that is in M to N o Remove the node M from the tree Algorithms for the Operations of the ADT Binary Search Tree Retrieval o Retrieval operation can be implemented by refining the search algorithm o Return the item with the desired search key if it exists o Otherwise, return a null reference 18
19 Algorithms for the Operations of the ADT Binary Search Tree traversal o Traversals for a binary search tree are the same as the traversals for a binary tree o Theorem 11-1 o The inorder traversal of a binary search tree T will visit its nodes in sorted search-key order A Reference-Based Implementation of the ADT Binary Search Tree BinarySearchTree o Extends BinaryTreeBasis o Inherits the following from BinaryTreeBasis o isempty() TreeIterator o makeempty() o getrootitem() o Can be used with BinarySearchTree 19
20 The Efficiency of Binary Search Tree Operations The maximum number of comparisons for a retrieval, insertion, or deletion is the height of the tree The maximum and minimum heights of a binary search tree o n is the maximum height of a binary tree with n nodes Theorem 11-2 A full binary tree of height h 0 has 2 h 1 nodes Theorem 11-3 The maximum number of nodes that a binary tree of height h can have is 2 h 1 20
21 Theorem 11-4 The minimum height of a binary tree with n nodes is log 2 (n+1) The height of a particular binary search tree depends on the order in which insertion and deletion operations are performed Treesort Uses the ADT binary search tree to sort an array of records into search-key order Efficiency o Average case: O(n * log n) o Worst case: O(n2) 21
22 HOMEWORK Read through Chapter 11 22
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