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Spring 2010CS 2251 Trees Chapter 6

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Spring 2010CS 2252 Chapter Objectives Learn to use a tree to represent a hierarchical organization of information Learn to use recursion to process trees To understand the different ways of traversing a tree To understand the difference between binary trees, binary search trees, and heaps To learn how to implement binary trees, binary search trees, and heaps using linked data structures and arrays Learn to use a binary search tree to store information for efficient retrieval Learn how to use a Huffman tree to encode characters compactly

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Spring 2010CS 2253 Trees Trees are nonlinear –useful for hierarchical organization of data Trees are recursive

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Spring 2010CS 2254 Tree Terminology A tree consists of a collection of elements or nodes, with each node linked to its successors The node at the top of a tree is called its root The links from a node to its successors are called branches The successors of a node are called its children The predecessor of a node is called its parent

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Spring 2010CS 2255 Tree Terminology Each node in a tree has exactly one parent except for the root node, which has no parent Nodes that have the same parent are siblings A node that has no children is called a leaf node A subtree of a node is a tree whose root is a child of that node

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Spring 2010CS 2256 Tree Terminology The level of a node is a measure of its distance from the root –If a node is the root, it has level 1 –Otherwise, a nodes level is one more than the level of its parent The height of a tree is the number of nodes in the longest path from the root to a leaf node –An empty tree has height 0 –The height of a tree is the same as the largest level that any node has

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Spring 2010CS 2257 Binary Trees In a binary tree, each node has at most two subtrees A set of nodes T is a binary tree if either of the following is true –T is empty –Its root node has two subtrees, TL and TR, such that TL and TR are binary trees

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Spring 2010CS 2258 Example 1:Expression tree Each node contains an operator or an operand –operator nodes have children –operand nodes are leaves

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Spring 2010CS 2259 Example 2: Huffman tree Represents Huffman codes for characters that might appear in a text file Huffman code uses different numbers of bits to encode letters as opposed to ASCII or Unicode

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Spring 2010CS 22510 Example 3: Binary Search Trees All elements in the left subtree precede those in the right subtree

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Spring 2010CS 22511 Fullness and Completeness A binary tree of height n is perfect if it has (2 n -1) nodes A binary tree is full if every node has two children except for the leaves A tree is complete if only the rightmost node(s) in the next to last row don't have two children

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Spring 2010CS 22512 General Trees Nodes of a general tree can have any number of subtrees A general tree can be represented using a binary tree

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Spring 2010CS 22513 Tree Traversals Often we want to determine the nodes of a tree and their relationship –Do this by walking through the tree in a prescribed order and visiting the nodes as they are encountered This process is called tree traversal –Itarators need to visit all nodes of a tree exactly once Three kinds of tree traversal –Inorder –Preorder –Postorder

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Spring 2010CS 22514 Preorder Tree Traversal Visit root node, traverse TL, traverse TR Algorithm if the tree is empty return else visit the root do preorder traversal of left subtree do preorder traversal of right subtree

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Spring 2010CS 22515 Inorder Tree Traversals Traverse TL, visit root node, traverse TR Algorithm if the tree is empty return else do preorder traversal of left subtree visit the root do preorder traversal of right subtree

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Spring 2010CS 22516 Postorder Tree Traversals Traverse TL, Traverse TR, visit root node Algorithm if the tree is empty return else do preorder traversal of left subtree do preorder traversal of right subtree visit the root

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Spring 2010CS 22517 Visualizing Tree Traversals You can visualize a tree traversal by imagining a mouse that walks along the edge of the tree –If the mouse always keeps the tree to the left, it will trace a route known as the Euler tour Preorder traversal if we record each node as the mouse first encounters it Inorder if each node is recorded as the mouse returns from traversing its left subtree Postorder if we record each node as the mouse last encounters it

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Spring 2010CS 22518 Visualizing Tree Traversals

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Spring 2010CS 22519 Traversals of Binary Search Trees An inorder traversal of a binary search tree results in the nodes being visited in sequence by increasing data value

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Spring 2010CS 22520 Traversals of Expression Trees An inorder traversal of an expression tree inserts parenthesis where they belong (infix form) A postorder traversal of an expression tree results in postfix form A preorder traversal of an expression results in prefix form

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Spring 2010CS 22521 Practice Traverse the following trees in all three orders

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Spring 2010CS 22522 Practice 2 Draw an expression tree whose inorder traversal is x / y + 3 * b / c Draw an expression tree whose postorder traversal is x y z + a b - c * / - Draw an expression tree whose preorder traversal is * + a - x y / c d

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Spring 2010CS 22523 Implementing trees Trees are generally implemented with linked structure similar to what we used for linked lists Nodes need to have references to at least two child nodes as well as the data A node may also have a parent reference.

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Spring 2010CS 22524 The Node Class Just as for a linked list, a node consists of a data part and links to successor nodes The data part is a reference to type E A binary tree node must have links to both its left and right subtrees

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Spring 2010CS 22525 The BinaryTree Class

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Spring 2010CS 22526 The BinaryTree Class

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Spring 2010CS 22527 Binary Search Trees Binary search tree definition –A set of nodes T is a binary search tree if either of the following is true T is empty Its root has two subtrees such that each is a binary search tree and the value in the root is greater than all values of the left subtree but less than all values in the right subtree

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Spring 2010CS 22528 Binary Search Tree http://www3.amherst.edu/~rjyanco94/literature/mothergoose/rhy mes/thisisthehousethatjackbuilt.html

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Spring 2010CS 22529 Searching a Binary Tree Searching for kept or jill

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Spring 2010CS 22530 Class Search Tree

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Spring 2010CS 22531 BinarySearchTree Class

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Spring 2010CS 22532 BinarySearchTreeData

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Spring 2010CS 22533 Binary Search Tree Insertion if the root is null create new node containing item to be the root else if item is the same as root data item is already in tree, return false else if item is less than root data search left subtree else search right subtree

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Spring 2010CS 22534 Binary Search Tree Delete if root is null return null else if item is less than root data return result of deleting from left subtree else if item is greater than root data return result of deleting from right subtree else // need to replace the root save data in root to return replace the root (see next slide)

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Spring 2010CS 22535 Replacing root of a subtree if root has no children set parent reference to local root to null else if root has one child set parent reference to root to child else // find the inorder predecessor if left child has no right child set parent reference to left child else find rightmost node in right child of left subtree and move its data to root

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Spring 2010CS 22536 Delete Example

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Spring 2010CS 22537 Practice Show the tree that would be created from the following items in the given order happy, depressed, manic, sad, ecstatic What happens if you add them in a different order? What happens if we add mad to the tree?

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Spring 2010CS 22538 Heaps and Priority Queues In a heap, the value in a node is les than all values in its two subtrees A heap is a complete binary tree with the following properties –The value in the root is the smallest item in the tree –Every subtree is a heap

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Spring 2010CS 22539 Inserting an Item into a Heap

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Spring 2010CS 22540 Removing from a Heap

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Spring 2010CS 22541 Implementing a Heap Because a heap is a complete binary tree, it can be implemented efficiently using an array instead of a linked data structure First element for storing a reference to the root data Use next two elements for storing the two children of the root Use elements with subscripts 3, 4, 5, and 6 for storing the four children of these two nodes and so on

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Spring 2010CS 22542 Computing Positions Parent of a node at position c is parent = (c - 1) / 2 Children of node at position p are leftchild = 2 p + 1 rightchild = 2 p + 2

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Spring 2010CS 22543 Inserting into a Heap Implemented as an ArrayList

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Spring 2010CS 22544 Removing from a Heap Implemented as an ArrayList

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Spring 2010CS 22545 Using Heaps The heap is not very useful as an ADT on its own –Will not create a Heap interface or code a class that implements it –Will incorporate its algorithms when we implement a priority queue class and Heapsort The heap is used to implement a special kind of queue called a priority queue

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Spring 2010CS 22546 Priority Queues Sometimes a FIFO queue may not be the best way to implement a waiting line –What if some entries have higher priority than others and need to be moved ahead in the line? A priority queue is a data structure in which only the highest-priority item is accessible

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Spring 2010CS 22547 Insertion into a Priority Queue Imagine a print queue that prints the shortest documents first

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Spring 2010CS 22548 The PriorityQueue Class Java provides a PriorityQueue class that implements the Queue interface given in Chapter 6. Peek, poll, and remove methods return the smallest item in the queue rather than the oldest item in the queue.

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Spring 2010CS 22549 Design of a KWPriorityQueue Class

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Spring 2010CS 22550 Huffman Trees A Huffman tree can be implemented using a binary tree and a PriorityQueue A straight binary encoding of an alphabet assigns a unique binary number to each symbol in the alphabet –Unicode for example The message “go eagles” requires 144 bits in Unicode but only 38 using Huffman coding

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Spring 2010CS 22551 Huffman Tree Example

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Spring 2010CS 22552 Huffman Trees

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Spring 2010CS 22553 Chapter Review A tree is a recursive, nonlinear data structure that is used to represent data that is organized as a hierarchy A binary tree is a collection of nodes with three components: a reference to a data object, a reference to a left subtree, and a reference to a right subtree In a binary tree used for arithmetic expressions, the root node should store the operator that is evaluated last

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Spring 2010CS 22554 Chapter Review A binary search tree is a tree in which the data stored in the left subtree of every node is less than the data stored in the root node, and the data stored in the right subtree is greater than the data stored in the root node A heap is a complete binary tree in which the data in each node is less than the data in both its subtrees Insertion and removal in a heap are both O(log n) A Huffman tree is a binary tree used to store a code that facilitates file compression

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