# Data Structures: Question Set – 26

#### Can a red-black tree have duplicate keys?

Yes, a red-black tree can have duplicate keys. When a node with a duplicate key is inserted into the tree, it is typically placed in a separate subtree to the right of the original node. When searching for a node with a specific key, the search may need to continue down the subtree of nodes with the same key until it finds the desired node.

#### What is the maximum height of a red-black tree with n nodes?

The maximum height of a red-black tree with n nodes is 2*log(n+1), where log is the logarithm base 2. This follows from the property that the longest path from the root to a leaf node in a red-black tree contains at most twice as many black nodes as the shortest path.

#### How does a red-black tree ensure that its height is balanced?

A red-black tree ensures that its height is balanced by maintaining the property that every path from the root to a leaf node contains the same number of black nodes. This property ensures that the longest path from the root to a leaf node contains at most twice as many black nodes as the shortest path, which ensures that the tree is roughly balanced.

#### Can a red-black tree have a node with both children being black?

Yes, a red-black tree can have a node with both children being black. This is not a violation of any of the properties of a red-black tree.

#### How does a red-black tree handle deletion of a node with two children?

When a node with two children is deleted from a red-black tree, it is replaced by its successor (the node with the smallest key in its right subtree) or its predecessor (the node with the largest key in its left subtree). If the replacement node is black, it may violate the red-black tree properties, so the tree is rebalanced by treating the replacement node as if it were a deleted node.

#### Can a red-black tree be used to implement a priority queue?

Yes, a red-black tree can be used to implement a priority queue. In this implementation, the keys in the tree represent the priorities of the elements in the queue, and the nodes are arranged in order of priority. Insertion and deletion operations can be performed in O(log n) time, which is optimal for a priority queue implemented using a binary search tree.

#### What is a graph?

A graph is a collection of vertices (also called nodes) and edges that connect the vertices. The edges represent relationships between the vertices.

#### What are some common applications of graphs?

Graphs have many applications, including:

- Modeling relationships between objects or people in social networks
- Representing transportation networks such as roads or flight routes
- Analyzing biological and chemical structures
- Optimizing supply chains and logistics
- Modeling computer networks
- Visualizing data in various fields such as finance, marketing, and engineering.

#### What is a directed graph?

A directed graph, also known as a digraph, is a graph where the edges have a direction. Each edge connects a source vertex to a target vertex, and the direction indicates the relationship between the two vertices.

#### What is an undirected graph?

An undirected graph is a graph where the edges have no direction. Each edge connects two vertices, and the relationship between the two vertices is symmetric.