paper.docx - Red and Black trees Abstract: This paper deals...

Red and Black trees
Abstract:
This paper deals with the concepts of red-black trees, which is one of the binary search tree. Each node of
the binary tree has an extra bit, and that bit is often interpreted as the color (red or black) of the node.
These color bits are used to ensure the tree remains approximately balanced during insertions and
deletions . From this paper I want to explain the behavior and properties of the red black trees by coming up
with problem statements in this method and their solutions.
1

Table of Contents:
 Introduction
 Properties
 Insertion
 Top down Red-Black trees
 Examples

Applications

Time Complexity
2


Conclusion
Introduction
:
A red–black tree is a special type of binary tree, used in computer science to organize pieces of comparable
data, such as text fragments or numbers. The leaf nodes of red–black trees do not contain data. These
leaves need not be explicit in computer memory—a null child pointer can encode the fact that this child is
a leaf—but it simplifies some algorithms for operating on red–black trees if the leaves really are explicit
nodes. To save memory, sometimes a single sentinel node performs the role of all leaf nodes; all references
from internal nodes to leaf nodes then point to the sentinel node. Red–black trees, like all binary search
trees, allow efficient in-order traversal (that is: in the order Left–Root–Right) of their elements. The search-
time results from the traversal from root to leaf, and therefore a balanced tree of n nodes, having the least
possible tree height, results in O(log n) search time. In an AVL tree, we recourse down the tree to find the
insertion (removal) location, then work back up the tree to update the heights and rebalance the tree. In a
red-black tree , a single pass down the tree will take care of business. An iterative solution is used.
Properties:
3

A red-black tree is a binary search tree with the following properties:
1. Every node is either red or black
2. Root node is black
3. All nodes are black
4. The children of any red node are black
5. Every path from a node to a null link must contain the same number of black nodes
These constraints enforce a critical property of red–black trees: the path from the root to the farthest leaf
is no more than twice as long as the path from the root to the nearest leaf. The result is that the tree is
roughly height-balanced. Since operations such as inserting, deleting, and finding values require worst-case
time proportional to the height of the tree, this theoretical upper bound on the height allows red–black
trees to be efficient in the worst case, unlike ordinary binary search trees.
To see why this is guaranteed, it suffices to consider the effect of properties 4 and 5 together. For a red–
black tree T, let B be the number of black nodes in property 5. Let the shortest possible path from the root
of T to any leaf consist of B black nodes. Longer possible paths may be constructed by inserting red nodes.