Question

For the following statements about red-black trees, provide a justification for each true statement and a counterexample for each false one. a. A subtree of a red-black tree is itself a red-black tree. b. The sibling of an external node is either external or it is red. c. There is a unique (2,4) tree associated with a given red-black tree. d. There is a unique red-black tree associated with a given (2,4) tree.

Short answer

a. True; b. False; c. False; d. True

Step-by-step solution

Analyze statement a

A subtree of a red-black tree is itself a red-black tree. This statement is true because the properties that define a red-black tree—every node is either red or black, the root is black, red nodes have black children, and every path from a given node to its descendant leaves has the same number of black nodes—are preserved in any subtree.

Analyze statement b

The sibling of an external node is either external or it is red. This statement is false. As a counterexample, consider a situation where the external node (null by definition) has a black sibling with red children. Hence, the sibling of an external node can be black.

Analyze statement c

There is a unique (2,4) tree associated with a given red-black tree. This statement is false. Different red-black trees can correspond to the same (2,4) tree due to the different ways red nodes can be arranged relative to their black parents in a red-black tree while still maintaining the (2,4) tree structure.

Analyze statement d

There is a unique red-black tree associated with a given (2,4) tree. This statement is true. Given a (2,4) tree, it maps uniquely to a red-black tree because each node in the (2,4) tree translates to a specific configuration of black and red nodes in the red-black tree preserving the red-black tree properties.

Key concepts

Red-Black Tree Properties

Red-black trees are a type of self-balancing binary search tree. They have several crucial properties which help maintain balance, making sure the longest path is no more than twice the shortest path from the root to any leaf. These properties are:

• Every node is either red or black.
• The root is always black.
• Red nodes cannot have red children (no two red nodes can be adjacent).
• Every path from a given node to its descendant leaves must have the same number of black nodes.

Understanding these properties is essential because they help in balancing the tree, ensuring logarithmic time complexity for search, insert, and delete operations. Essentially, these properties are maintained during any operation (inserts or deletes) to ensure the tree remains balanced.

Tree Data Structures

Tree data structures are a way of organizing and storing data hierarchically. Trees consist of nodes connected by edges. Each node in a tree has a value and children nodes.
One of the fundamental types of tree data structures is the binary tree, where each node has at most two children, typically referred to as the left and right child. More advanced forms include balanced trees like AVL trees, B-trees, and of course, red-black trees.
Trees are especially useful for scenarios where you need to maintain a hierarchical structure or perform frequent insertions and deletions. The balanced nature of red-black trees, for instance, ensures efficient operations, making them suitable for implementing dictionaries and priority queues.

Binary Search Trees

Binary search trees (BSTs) are a specific category of binary trees that facilitate efficient searching, insertion, and deletion operations.
In a BST, for each node:

• The left subtree contains only nodes with values less than the node's value.
• The right subtree contains only nodes with values greater than the node's value.
• Both left and right subtrees are also BSTs.

Thus, BSTs permit fast lookup, addition, and removal of values maintaining average time complexity of O(log n). However, unbalanced BSTs could degrade to O(n) time complexity, which is why balanced variations like red-black trees are preferable in practice.

(2,4) Trees

(2,4) trees are a type of multi-way balanced search tree used primarily in database and file systems to maintain ordered data.
This data structure allows nodes to have between 2 and 4 children, balancing the tree by ensuring that the path lengths from the root to any leaf node are almost the same. This characteristic guarantees logarithmic height, consequently optimizing search operations.
Each internal node of a (2,4) tree can store between 1 to 3 keys. Red-black trees can be seen as binary tree representations of (2,4) trees, where:

• A single black node corresponds to one key in the (2,4) tree.
• A red-black pair (a red node with a black parent) represents an internal path within a node of the (2,4) tree.

The relationship between red-black and (2,4) trees ensures efficient performance in terms of balancing and operations.