Self-Balancing-Binary-Search-Trees (Comparisons)

Self-Balancing Binary Search Trees are height-balanced binary search trees that automatically keeps height as small as possible when insertion and deletion operations are performed on tree. The height is typically maintained in order of Log n so that all operations take O(Log n) time on average.

Examples :
Red Black Tree
Red Black Tree

AVL Tree:

How do Self-Balancing-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.

We have already discussed AVL tree, Red Black Tree and Splay Tree. In this acrticle, we will compare the efficiency of these trees:

Metric RB Tree AVL Tree Splay Tree
Insertion in
worst case
O(1) 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 Multiset, Multimap, Map, Set, etc. Database Transactions Cache implementation, Garbage collection Algorithms


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