Exploring Trees and Heaps

What is the maximum number of children a node can have in a B-tree of order m?

• 2m

• m-1

• m

• m/2

Describe the properties that define a B-tree.

• A B-tree is a balanced tree with nodes containing multiple keys and children, all leaf nodes at the same level, and efficient operations for insertion, deletion, and search.

• A B-tree has all leaf nodes at different levels for better performance.

• A B-tree is an unbalanced tree that allows only sequential access.

• A B-tree is a linear structure with a single key per node.

What is the time complexity of searching for a value in a B-tree?

• O(n)

• O(n log n)

• O(1)

• O(log n)

Explain the process of left rotation in a red-black tree.

• Left rotation is a tree restructuring operation that moves a node down to the left and its right child up to its place.

• Left rotation is a process that only applies to binary search trees, not red-black trees.

• Left rotation involves moving a node up to the right and its left child down to its place.

• Left rotation is a method to delete a node from the tree without restructuring.

What are the color properties that must be maintained in a red-black tree?

• Every node must have at least one red child.

• The root can be either red or black.

• All nodes must be red.

• The color properties of a red-black tree are: 1. Each node is red or black. 2. The root is black. 3. Red nodes cannot have red children. 4. Every path from a node to its leaves has the same number of black nodes. 5. All leaves are black.

How does a red-black tree ensure balance after insertion?

• Red-black trees ensure balance after insertion by using rotations and recoloring to fix violations of red-black properties.

• By only allowing nodes with two children.

• By ignoring the color properties during insertion.

• By deleting the node immediately after insertion.

What is the time complexity of inserting an element into a red-black tree?

• O(1)

• O(n)

• O(log n)

• O(n log n)

Describe the merging process of two binomial heaps.

• The merging process consists of removing all trees from both heaps before combining them.

• The merging process involves sorting the elements in ascending order.

• The merging process requires creating a new binomial heap from scratch.

• The merging process involves combining root lists and linking trees of the same degree.

What is the time complexity of merging two binomial heaps?

• O(n log n)

• O(log n)

• O(n)

• O(1)

How do you perform a search operation in a binomial heap?

• Traverse each binomial tree and perform a linear search for the desired element.

• Use a priority queue to find the element.

• Perform a binary search on the binomial heap.

• Directly access the root of the binomial heap for the element.