Self-Balancing Binary Search Trees

Self-Balancing Binary Search Trees are height-balanced binary search trees that automatically keep the height as small as possible when insertion and deletion operations are performed on the tree. 

The height is typically maintained in order of logN so that all operations take O(logN) time on average. 

Examples: The most common examples of self-balancing binary search trees are 

• AVL Tree
• Red Black Tree and 
• Splay Tree

AVL Tree:

An AVL tree defined as a self-balancing Binary Search Tree (BST) where the difference between heights of left and right subtrees for any node cannot be more than one.

Example of AVL Tree

Basic operations on AVL Tree include:

• Insertion
• Searching
• Deletion

To learn more about this, refer to the article on AVL Tree.

Red-Black Tree:

Red-Black tree is a self-balancing binary search tree in which every node is colored with either red or black. The root and leaf nodes (i.e., NULL nodes) are always marked as black.

Example of Red-Black Tree

Some Properties of Red-Black Tree:

• Root property: The root is black.
• External property: Every leaf (Leaf is a NULL child of a node) is black in Red-Black tree.
• Internal property: The children of a red node are black. Hence possible parent of red node is a black node.
• Depth property: All the leaves have the same black depth.
• Path property: Every simple path from the root to descendant leaf node contains the same number of black nodes

To learn more about this, refer to the article on “Red-Black Tree“.

Splay Tree:

Splay is a self-balancing binary search tree. The basic idea behind splay trees is to bring the most recently accessed or inserted element to the root of the tree by performing a sequence of tree rotations, called splaying.

Basic operations that are performed in a splay tree are:

• Insertion
• Searching
• Deletion
• Rotation (There are two types of rotation in a Splay tree named zig rotation and zag rotation)

To learn more about this, refer to the article on “Splay Tree“.

Language Implementations of self-balancing BST:

• set and map in C++ STL. 
• TreeSet and TreeMap in Java. Most of the library implementations use Red Black Tree. 
• Python standard library does not support Self Balancing BST. In Python, we can use bisect module to keep a set of sorted data. We can also use PyPi modules like rbtree (implementation of red-black tree) and pyavl (implementation of AVL tree).

How does Self-Balancing Binary Search Tree maintain height?

A typical operation done by trees is rotation. Following are two basic operations that can be performed to re-balance a BST without violating the BST property (keys(left) < key(root) < keys(right)). 

1. Left Rotation 
2. Right Rotation  

T1, T2 and T3 are subtrees of the tree rooted with y (on the left side) or x (on the right side)           

          y                                                 x
        /  \        Right Rotation         /  \
      x   T3      – – – – – – – – – >          T1   y 
   /   \           <- – – – – – – – –          /   \
T1  T2         Left Rotation   T2  T3

Keys in both of the above trees follow the following order 

keys(T1) < key(x) < keys(T2) < key(y) < keys(T3)

So BST property is not violated anywhere.

Comparisons among Red-Black Tree, AVL Tree and Splay Tree:

In this article, we will compare the efficiency of these trees:

Metric	Red-Black Tree	AVL Tree	Splay Tree
Insertion in  worst case	O(logN)	O(logN)	Amortized O(logN)
Maximum height  of tree	2*log(n)	1.44*log(n)	O(n)
Search in  worst case	O(logN),  Moderate	O(logN),  Faster	Amortized O(logN),  Slower
Efficient Implementation requires	Three pointers with color bit per node	Two pointers with balance factor per  node	Only two pointers with  no extra information
Deletion in  worst case	O(logN)	O(logN)	Amortized O(logN)
Mostly used	As universal data structure	When frequent lookups are required	When same element is  retrieved again and again
Real world Application	Database Transactions	Multiset, Multimap, Map, Set, etc.	Cache implementation, Garbage collection Algorithms

Conclusion:

AVL Tree is considered to be Strict as we have to maintain Balance Factor i.e. abs(Height of Left Sub-Tree – Height of Right Sub-Tree) ≤ 1, and In Red Black Tree, it is not considered to be as Strict as AVL Tree, due to which there are less number of rotations are required to make the Unbalanced BST -> Balanced as compared to AVL Tree. Also, In summary, if the application requires fast search times, an AVL Tree may be the better choice. If the application requires fast insertion and deletion times, a Red-Black Tree may be the better choice. It’s also important to consider other factors such as the size and complexity of the data being stored, as well as the memory and processing resources available.