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Binary Tree and General Tree. Chapter 9. Overview. Two-way decision making is one of the fundamental concepts in computing. A binary tree models two-way decisions. A hierarchy represents multi-way choices. The general tree is an extension of the binary tree. Learning Objectives.
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Binary Tree and General Tree Chapter 9
Overview • Two-way decision making is one of the fundamental concepts in computing. • A binary tree models two-way decisions. • A hierarchy represents multi-way choices. • The general tree is an extension of the binary tree.
Learning Objectives • Describe a binary tree in terms of its structure and components, and learn recursive definitions of the binary tree and its properties. • Study standard tree traversals in depth. • Develop a binary tree class interface based on its recursive definition. • Learn about the signature of a binary tree and understand how to build a binary tree given its signature.
Learning Objectives • Understand Huffman coding, a binary tree-based text compression application, and use the binary tree class to implement Huffman coding. • Implement the binary tree class. • Study how tree traversals may be implemented non-recursively using a stack. • Describe the properties of a general tree.
Learning Objectives • Learn the natural correspondence of a general tree with an equivalent binary tree, and the signature of a general tree.
9.1.1 Components • A binary tree consists of nodes and branches. • A node is a place in the tree where data is stored. • There is a special node called the root. • “starting point” of the tree. • The nodes are connected to each other by links or branches. • A left branch or right branch. • Binary means that there are at most two choices. • A node is said to have at most two children. • A node that does not have any children is called a leaf. • Non-leaf nodes are called internal nodes.
9.1.1 Components • There is a single path from any node to any other node in the tree.
9.1.2 Position as Meaning • If-then-else tree • Represents an if-then-else construct in a program. • Every node in this tree is conditional expression that evaluates to yes or no. • If it evaluates to yes, the left branch (if any) is taken and if evaluates to no, the right branch (if any) is taken.
9.1.2 Position as Meaning • Expression tree • (f + ((a * b) - c)) Double-click to add graphics
Create Expression tree • 1- If the current token is a '(', add a new node as the left child of the current node, and descend to the left child. • 2- If the current token is in the list ['+','-','/','*'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child. • 3- If the current token is a number, set the root value of the current node to the number and return to the parent. • 4- If the current token is a ')', go to the parent of the current node.
let’s look at an example of the rules outlined above in action. We will use the expression (3+(4∗5)). • We will parse this expression into the following list of character tokens['(', '3', '+', '(', '4', '*', '5' ,')',')']. • Initially we will start out with a parse tree that consists of root node.
9.1.3 Structure • Structure • Two trees with the same number of nodes may not have the same structure.
9.1.3 Structure • Depth is the distance from the root. • Nodes at the same depth are said to be at the same level, with the root being at level zero. • The height of a tree is the maximum level (or depth) at which there is a node.
9.1.3 Structure • Full Binary Tree: A binary tree in which all of the leaves are on the same level and every nonleaf node has two children
9.1.3 Structure • (a), first three are strictly binary, but the fourth is not. first two are FULL binary tree • (b), first two are complete and the last two are not. • At level i, there can be at most 2i nodes. • Maximum number of nodes over all the levels.
Traversal Definitions • Preorder traversal: Visit the root, visit the left subtree, visit the right subtree • Inorder traversal: Visit the left subtree, visit the root, visit the right subtree • Postorder traversal: Visit the left subtree, visit the right subtree, visit the root
Overview • A binary tree possesses ordering property that maintains the data in its nodes in sorted order. • Since the search tree is a linked structure, entries may be inserted and deleted without having to move other entries over, unlike ordered lists in which insertions and deletions require data movement. • The AVL tree is a height-balanced binary search tree that delivers guaranteed worst-case search, insert, and delete times that are all O(log n)
Learning Objectives • Explore the motivation for binary search trees by learning about the comparison tree for binary search. • Use the comparison tree as an analytical tool to determine the running time of binary search. • Describe the binary search tree structure and properties. • Study the primary binary search tree operations of search, insert, and delete, and analyze their running times.
Learning Objectives • Understand a binary search tree class interface and use it in application examples. • Implement the binary search tree class with a binary tree class as the reused storage component. • Study the AVL tree structure properties, the search, insert, and delete operations, and their running times.
10.2 Binary Search Tree Properties • All three trees have the same set of keys. • Their structures are different, depending on the sequence of insertion or deletion.
10.3 Binary Search Tree Operations • Three foundational operations • Search • Insert • Delete
10.3.1 Search • The tree nodes are for real. • The target key is compared against the key at the root of the tree. • If they are equal, sucess. • If not, recusively search the appropriate child. • Search terminates with failure if an empty subtree is reached.
10.3.2 Insert • To insert a value, search must force a failure. • Item in inserted in the failed location. • A newly inserted node always becomes a leaf node in the search tree.
10.3.3 Delete • The value to be deleted is first located in the binary search tree. • Three possible cases. • Case a: X is a leaf node.
10.3.3 Delete • Case b: X has one child • Replace the deleted node with the child.
10.3.3 Delete • Case c: X has two children • Find the inorder predecessor, Y, of X. • Copy the entry at Y into X. • Apply deletion on Y. • Applying deletion on Y will revert to either case b or a since Y is guaranteed to not have a right subtree.
Running Times • Search: worst case • Tied to the worst possible shape a tree can attain. • Such a tree degenerates into sequential search. • O(n).
Running Times • Insertion: worst case • O(n) • Deletion: worst case • O(n)
Balancing • Keeping a binary search tree balanced allows the height never to exceed O(log n). • There are two popular ways of maintaining and constructing balanced binary search trees. • AVL tree. • red-black tree.
10.5.1 Example Treesort • inOrder traversal method invokes visitor.visit() when a node is visited.
10.5.2 Example: Counting Keys • Count the number of keys in a binary search tree that are less than a given key. • It is possible to simply examine every node of the tree.
