- How well-versed are you with the concepts of AVL tree?
- Have you tried to go inside its detailing?
- How to implement this concept in binary search trees?
If you too are looking for the answers of all the above questions, this tutorial is perfect for you!
Touted as the self-balancing trees, AVL trees are helpful in restoring the balance in case there will be an imbalance while the insertion or the deletion of binary tree nodes.
The whole key lies in the balance and AVL trees very well know how to maintain this balance!
Though, deletion in AVL trees is a far-fetched concept. So, to get inside the better detailing of it, stay with us till the end of this blog.
Let’s get started!
AVL Tree
Adelson Velskii or Landis [ AVL] is defined as the binary search tree where there is a difference between the right or left subtree at the 1. This difference is also defined by the name of the balancing factor.
A binary tree is touted to be the AVL tree if it satisfies the below-mentioned condition
- This will be the binary search tree
- For a given node A, the height of your left subtree would also be A which will differ by
Deletion in case of AVL tree
Do you know how deletion in the AVL tree occurs?
Well, when we are inserting or deleting any element in the tree, its structure will change. To restore the properties of AVL, we should modify the tree. Well, this can be done with the help of rotations.
Though, the insertion or deletion will either add or delete the single node. This will only increase or decrease a subtree’s height by 1. After performing all the operations, the AVL tree will get restored to its single or the double rotation.
Algorithm
This self-balancing tree will delete all the nodes in the same way as it is being done with the help of a binary tree. Firstly we will check for the deleted tree, then we will check for the unbalanced nodes of the tree.
The remaining steps will be as follows:
- After the nodes of the tree get deleted, start from the parent node and move till the roots of the tree. Firstly check if there will be any unbalanced node present in the tree
- Restore all the properties of the AVL tree by performing all the remaining rotations
Performing the operations
To perform the required operations in the deletion of AVL tree, let’s first consider an example
A will be the critical node while B will be your root node. If the node X will be there in your left subtree as A, we have to do it by taking into consideration, three different kind of situations:
R0 Rotation [ Here the node B will have the balancing factor 0]
In this case, node B would be having the balancing factor with 0 value, the balancing factor of the node A will be distributed as the deletion of node X with the rebalancing of the tree using R0 rotation.
Here the given critical node A will be moved to the right while the node B would be the root node with T1 as your left sub tree. T2 and T3 will be the left or right subtrees of the node A.
Here you can rotate the node having balancing factor 0 to get your actual node. The right node will become the left child of the node A
R1 Rotation [ Here the node B will be having the balancing factor 1]
R1 rotation needs to be performed if their balancing factor would be 1. Here, the critical node A gets moved to T2 which is a left subtree while T3 would be the right subtree. T1 will also be placed as the left subtree in this case.
After deleting the disturbed nodes from your tree, you will get the critical node. In this way, you will get the R1 condition, where node A gets shifted to the right and the right of B will become the left of A. Hence, you will get the solution.
R-1 Rotation [ Here, the node B will be having the balancing factor -1
R-1 rotation will be performed in case the node B will be having the balancing factor as -1. We will treat it the same way as the LR rotation. Here, the node C which is the right child of the B node will become the root node for the right and left children respectively.
When the node B will be having the balancing factor as -1, you will delete all of the remaining nodes which will disturb your balancing factor.
Advantages of AVL Tree
The concept of AVL tree comes with a lot of benefits that every user should know:
- AVL tree are considered as the balanced one so they are more balanced than the general tree to the binary tree
- The main advantage of these trees is that they can be performed in a better way as compared to the binary search trees or red black trees especially when it is related with the search operations
- These trees are considered better in comparison to the other trees. Moreover, in terms of time complexity they do give better results than the binary search tree
- Low time complexity will also be needed to add or delete the nodes in a given tree structure
Can you delete the root node in the AVL tree?
Well, this is one of the commonly asked questions. The answer will be a sureshot yes as you will be able to delete all the root nodes in your AVL tree. For that, you need to find nodes of the in-order successor from the root node.
Wrapping Up
Keep this tutorial as your mentor to learn the knits and grits of deletion in the AVL tree!
You can also equip yourself with other tree related concepts like converting general tree to binary tree and much more!
