Binary Trees,
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05102010, 05:28 PM
bst3.ppt (Size: 1.64 MB / Downloads: 128) Binary Trees, an d Binary Search Trees and Binary Search Trees Linear access time of linked lists is prohibitive Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? Trees Basic concepts Tree traversal Binary tree Binary search tree and its operations 


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20012011, 12:42 PM
Binary Search Trees.PPT (Size: 360 KB / Downloads: 40) Binary Search Trees Binary Search Trees One of the tree applications in Chapter 10 is binary search trees. In Chapter 10, binary search trees are used to implement bags and sets. This presentation illustrates how another data type called a dictionary is implemented with binary search trees. The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key. The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key. The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key. The Dictionary Data Type The insertion procedure for a dictionary has two parameters. The Dictionary Data Type When you want to retrieve an item, you specify the key... The Dictionary Data Type The Dictionary Data Type We'll look at how a binary tree can be used as the internal storage mechanism for the dictionary. A Binary Search Tree of States The data in the dictionary will be stored in a binary tree, with each node containing an item and a key. A Binary Search Tree of States Storage rules: Every key to the left of a node is alphabetically before the key of the node. A Binary Search Tree of States Storage rules: Every key to the left of a node is alphabetically before the key of the node. A Binary Search Tree of States Storage rules: Every key to the left of a node is alphabetically before the key of the node. Every key to the right of a node is alphabetically after the key of the node. A Binary Search Tree of States Storage rules: Every key to the left of a node is alphabetically before the key of the node. Every key to the right of a node is alphabetically after the key of the node. Retrieving Data Start at the root. If the current node has the key, then stop and retrieve the data. If the current node's key is too large, move left and repeat 13. If the current node's key is too small, move right and repeat 13. Retrieve ' New Hampshire' Start at the root. If the current node has the key, then stop and retrieve the data. If the current node's key is too large, move left and repeat 13. If the current node's key is too small, move right and repeat 13. Retrieve 'New Hampshire' Start at the root. If the current node has the key, then stop and retrieve the data. If the current node's key is too large, move left and repeat 13. If the current node's key is too small, move right and repeat 13. Retrieve 'New Hampshire' Start at the root. If the current node has the key, then stop and retrieve the data. If the current node's key is too large, move left and repeat 13. If the current node's key is too small, move right and repeat 13. Retrieve 'New Hampshire' Start at the root. If the current node has the key, then stop and retrieve the data. If the current node's key is too large, move left and repeat 13. If the current node's key is too small, move right and repeat 13. Adding a New Item with aGiven Key Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Pretend that you are trying to find the key, but stop when there is no node to move to. Add the new node at the spot where you would have moved to if there had been a node. Adding Adding Removing an Item with a Given Key Find the item. If necessary, swap the item with one that is easier to remove. Remove the item. Removing 'Florida' Find the item. Removing 'Florida' Removing 'Florida' Removing 'Florida' If necessary, do some rearranging. Removing 'Florida' If necessary, do some rearranging. Removing 'Florida' If necessary, do some rearranging. Removing 'Florida' If necessary, do some rearranging. Removing 'Florida' If necessary, do some rearranging. Removing 'Florida' Removing 'Florida' Removing an Item with a Given Key Find the item. If the item has a right child, rearrange the tree: Find smallest item in the right subtree Copy that smallest item onto the one that you want to remove Remove the extra copy of the smallest item (making sure that you keep the tree connected) else just remove the item. Summary Binary search trees are a good implementation of data types such as sets, bags, and dictionaries. Searching for an item is generally quick since you move from the root to the item, without looking at many other items. Adding and deleting items is also quick. 


seminar ideas Super Moderator Posts: 10,003 Joined: Apr 2012 
13042012, 03:08 PM
Binary Trees
BinaryTrees.pdf (Size: 49.74 KB / Downloads: 14) Stanford CS Education Library  #110 This is article #110 in the Stanford CS Education Library. This and other free CS materials are available at the library (cslibrary.stanford.edu/). That people seeking education should have the opportunity to find it. This article may be used, reproduced, excerpted, or sold so long as this paragraph is clearly reproduced. Copyright 20002001, Nick Parlante, nick.parlante@cs.stanford.edu. Introduction To Binary Trees A binary tree is made of nodes, where each node contains a "left" pointer, a "right" pointer, and a data element. The "root" pointer points to the topmost node in the tree. The left and right pointers recursively point to smaller "subtrees" on either side. A null pointer represents a binary tree with no elements  the empty tree. The formal recursive definition is: a binary tree is either empty (represented by a null pointer), or is made of a single node, where the left and right pointers (recursive definition ahead) each point to a binary tree. Binary Search Tree Niche Basically, binary search trees are fast at insert and lookup. The next section presents the code for these two algorithms. On average, a binary search tree algorithm can locate a node in an N node tree in order lg(N) time (log base 2). Therefore, binary search trees are good for "dictionary" problems where the code inserts and looks up information indexed by some key. The lg(N) behavior is the average case  it's possible for a particular tree to be much slower depending on its shape. Strategy Some of the problems in this article use plain binary trees, and some use binary search trees. In any case, the problems concentrate on the combination of pointers and recursion. (See the articles linked above for pointer articles that do not emphasize recursion.) For each problem, there are two things to understand... The node/pointer structure that makes up the tree and the code that manipulates it The algorithm, typically recursive, that iterates over the tree When thinking about a binary tree problem, it's often a good idea to draw a few little trees to think about the various cases. Pointer Changing Code There is a common problem with pointer intensive code: what if a function needs to change one of the pointer parameters passed to it? For example, the insert() function below may want to change the root pointer. In C and C++, one solution uses pointerstopointers (aka "reference parameters"). That's a fine technique, but here we will use the simpler technique that a function that wishes to change a pointer passed to it will return the new value of the pointer to the caller. Binary Tree Problems Here are 14 binary tree problems in increasing order of difficulty. Some of the problems operate on binary search trees (aka "ordered binary trees") while others work on plain binary trees with no special ordering. The next section, Section 3, shows the solution code in C/C++. Section 4 gives the background and solution code in Java. The basic structure and recursion of the solution code is the same in both languages  the differences are superficial. Java Binary Trees and Solutions In Java, the key points in the recursion are exactly the same as in C or C++. In fact, I created the Java solutions by just copying the C solutions, and then making the syntactic changes. The recursion is the same, however the outer structure is slightly different. 


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12112012, 12:08 PM
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