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C++ Program to Implement AVL Tree
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Your browser does not support inline frames or is currently configured not to display inline frames.This article assumes knowledge of the binary search tree BST data structure. There are two main ways of representing a binary tree. The first is using node objects that have references to their children. The second is using a regular array and manipulating the index of the node to find its children.
The height of a node is defined as the maximum number of steps from the node to any of its leaf nodes. When insertion or deletion occurs, the heights of nodes are updated and a balancing operation occurs. There are four different cases that can occur, each of them can be fixed using either one or two binary tree rotation operations. It can be fixed using a single right rotate operation on the unbalanced node. It can be fixed using a left rotate on the left child, which results in the left left case which is fixed using a right rotate on the unbalanced node.
This is the inverse of the left left case. It can be fixed using a single left rotate operation on the unbalanced node. This is the inverse of the left right case. It can be fixed using a right rotate on the right child, which results in the right right case which is fixed using a left rotate on the unbalanced node. Delete starts by performing a regular BST delete. Once the node is deleted, go up the tree updating the heights of each node and rebalancing when required as outlined in the balancing section.
Insert starts by performing a regular BST insert. Once the node is inserted, go up the tree updating the heights of each node and rebalancing when required as outlined in the balancing section.
The worst case balancing of the AVL tree occurs when one child on every node has a height of 1 more than the other child. Notice how the example of a tree of height 3 is a node with the left child being the height 2 tree and right child being the height 1 tree. This illustrates the recurrence for the worst case height of a tree. Fortunately, just like the Fibonacci sequence, the number of nodes in a worst case balanced AVL tree increases exponentially.
The AVL tree, red-black tree and splay tree are all self-adjusting binary search trees, so which one is better in which situation? And when is it better to use a regular BST? A paper by Ben Pfaff of Stanford University performs an in-depth study of the performance characteristics of each tree under various circumstances.
Each data structure excels based on runtime patterns in the input and the calling of operations. It comes to the following conclusions:. Tree representation The second is using a regular array and manipulating the index of the node to find its children.What if the input to binary search tree comes in a sorted ascending or descending manner? In real-time data, we cannot predict data pattern and their frequencies.
So, a need arises to balance out the existing BST. AVL tree checks the height of the left and the right sub-trees and assures that the difference is not more than 1. This difference is called the Balance Factor. In the second tree, the left subtree of C has height 2 and the right subtree has height 0, so the difference is 2. In the third tree, the right subtree of A has height 2 and the left is missing, so it is 0, and the difference is 2 again. AVL tree permits difference balance factor to be only 1.AVL tree insertion
If the difference in the height of left and right sub-trees is more than 1, the tree is balanced using some rotation techniques. The first two rotations are single rotations and the next two rotations are double rotations. To have an unbalanced tree, we at least need a tree of height 2.
With this simple tree, let's understand them one by one. In our example, node A has become unbalanced as a node is inserted in the right subtree of A's right subtree. We perform the left rotation by making A the left-subtree of B. AVL tree may become unbalanced, if a node is inserted in the left subtree of the left subtree. The tree then needs a right rotation. As depicted, the unbalanced node becomes the right child of its left child by performing a right rotation.
Double rotations are slightly complex version of already explained versions of rotations. To understand them better, we should take note of each action performed while rotation.
Have you got any ideas? Up to this moment I have the following tests:. Russian wikipedia says that it is proven experimentally, that for two insertions one rebalancing needed and for five removals also one rebalancing needed, but is it really so? English wikipedia says nothing about it, and for my AVL one rebalancing needed for two insertions or for four removals, which is not quite the same.
Maybe these tests are enough, but if there are any more tests, not difficult to implement, why not to do it? That means that covering the basic scenarios should give you a broad coverage of the AVL tree functionality. In other words, these tests done on the smallest tree structure that allows them are the most important ones:.
If your implementation passes these tests, it would probably pass them on larger trees. Note that performance and memory usage is not tested here. In the spirit of all these answers, I thought I'd provide a couple concrete examples to demonstrate that the basic case is not enough. To wit, the final landing place of the inserted value and the final balance factors of node 4 and node 20 are entirely dependent on the relative value of the right child under node 4 - if any.
A test solely off any one of these cases does not necessarily prove the correctness of any others. Note: node 4 must initially be balanced for these cases; an initial imbalance in node 4 ultimately has no effect on node The more complex cases were a problem for me when I was working on optimizing the calculation of balance factors that is, adjusting balance factors only for affected nodes rather than recalculating the entire tree. Delete node 1 from each of these trees.
Note that Case 1 effectively proves Case 2, but not at all Case 3. The are plenty of examples of AVL rotations in books and on the internet, but what I found seemed arbitrary and no one place seemed to include simple examples for all 4 cases for insert and delete.
These are the simplest test case I could come up with for the 4 kinds of rotations. For example, the string "abc" would be insert "a", insert "b" and then insert "c". The first causes the rotations, but has empty sub-trees for the nodes being rotated making it easy to see what actually happened. The second suite has non-empty sub-trees to fully test the rotation code.Remarks : By default, we show e-Lecture Mode for first time or non logged-in visitor.
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A Table ADT must support at least the following three operations as efficient as possible:. Reference: See similar slide in Hash Table e-Lecture. Another pro-tip: We designed this visualization and this e-Lecture mode to look good on x resolution or larger typical modern laptop resolution in We recommend using Google Chrome to access VisuAlgo. Go to full screen mode F11 to enjoy this setup. Drop an email to visualgo. It has very fast Search vInsert vand Remove v performance all in expected O 1 time.
Root vertex does not have a parent. There can only be one root vertex in a BST. Leaf vertex does not have any child. There can be more than one leaf vertex in a BST. Vertices that are not leaf are called the internal vertices. Sometimes root vertex is not included as part of the definition of internal vertex as the root of a BST with only one vertex can actually fit into the definition of a leaf too. Not all attributes will be used for all vertices, e. Some other implementation separates key for ordering of vertices in the BST with the actual satellite data associated with the keys.
The parent of a vertex except root is drawn above that vertex. The integer key of each vertex is drawn inside the circle that represent that vertex. In the example above, key 15 has 6 as its left child and 23 as its right child. Thus the parent of 6 and 23 is Before going through this, please read the following two articles that discuss about the binary search tree and self-balancing binary search tree. AVL trees are height balanced binary search trees. AVL tree keeps the height balanced using the following property.
The heights of the left and right subtrees differ by at most 1. Every node should follow the above property and the resulting tree is the AVL tree. If any of the node violates this property, the tree should be re-balanced to maintain the property.
Consider an AVL tree given in Figure 1. The total number of nodes in the tree is the sum of the total number of nodes in the left subtree, the total number of nodes in the right subtree and the root node. In an AVL tree, the balance factor must be -1, 0, or 1. If the balance factor of a node is greater than 1 right heavy or less than -1 left heavythe node needs to be rebalanced.
Figure 2 shows a tree with balance factor. When the balance factor of a node is less than -1 or greater than 1, we perform tree rotations on the node. These rotations change the structure of the tree and make the tree balanced.
There are four kind of rotations we do in the AVL tree. These are described below. The tree at the left side of Figure 3 is right heavy. This is done in the following steps. Sometimes a single rotation is not sufficient to balance an unbalanced tree.
Consider a tree given in Figure 5. As we can see in the figure, the tree after the left rotation is still un-balanced. Therefore the single left operation is not effective in this case. To fix this we do the following using two step rotation.AVL tree is a self-balancing Binary Search Tree BST where the difference between heights of left and right subtrees cannot be more than one for all nodes.
Why AVL Trees?
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Most of the BST operations e. The cost of these operations may become O n for a skewed Binary tree. If we make sure that height of the tree remains O Logn after every insertion and deletion, then we can guarantee an upper bound of O Logn for all these operations. The height of an AVL tree is always O Logn where n is the number of nodes in the tree See this video lecture for proof. Insertion To make sure that the given tree remains AVL after every insertion, we must augment the standard BST insert operation to perform some re-balancing.
Let z be the first unbalanced node, y be the child of z that comes on the path from w to z and x be the grandchild of z that comes on the path from w to z. There can be 4 possible cases that needs to be handled as x, y and z can be arranged in 4 ways. Following are the possible 4 arrangements: a y is left child of z and x is left child of y Left Left Case b y is left child of z and x is right child of y Left Right Case c y is right child of z and x is right child of y Right Right Case d y is right child of z and x is left child of y Right Left Case.
Following are the operations to be performed in above mentioned 4 cases. In all of the cases, we only need to re-balance the subtree rooted with z and the complete tree becomes balanced as the height of subtree After appropriate rotations rooted with z becomes same as it was before insertion.
See this video lecture for proof. Insertion Examples:. The following implementation uses the recursive BST insert to insert a new node. In the recursive BST insert, after insertion, we get pointers to all ancestors one by one in a bottom-up manner. The recursive code itself travels up and visits all the ancestors of the newly inserted node.
Update the height of the current node. To check whether it is left left case or not, compare the newly inserted key with the key in left subtree root. To check whether it is Right Right case or not, compare the newly inserted key with the key in right subtree root. Time Complexity: The rotation operations left and right rotate take constant time as only a few pointers are being changed there. Updating the height and getting the balance factor also takes constant time. The AVL trees are more balanced compared to Red-Black Trees, but they may cause more rotations during insertion and deletion.
So if your application involves many frequent insertions and deletions, then Red Black trees should be preferred. And if the insertions and deletions are less frequent and search is the more frequent operation, then AVL tree should be preferred over Red Black Tree. Following is the post for delete. Median in a stream of integers running integers Maximum of all subarrays of size k Count smaller elements on right side.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Attention reader!
Writing code in comment? Please use ide. NULL left and right pointers. Get the balance factor of this ancestor. Node left, right. Node root. Python code to insert a node in AVL tree. AVL tree class which supports the.