How is a red-black tree balanced in data structures?

A red-black tree is balanced through a process of rotations and colour changes during insertion and deletion operations.

A red-black tree is a type of self-balancing binary search tree, where each node has an extra attribute: colour, which can be either red or black. The balancing of the tree is maintained by applying a set of rules, known as red-black properties, which are enforced during the insertion and deletion operations.

The red-black properties are as follows:
1. Every node is either red or black.
2. The root node is always black.
3. All leaves (null or NIL nodes) are black.
4. If a node is red, then both its children are black.
5. Every path from a node to its descendant leaves contains the same number of black nodes.

When a new node is inserted into the tree or an existing node is deleted, these operations could potentially violate the red-black properties. To restore the balance and ensure the properties are still satisfied, the tree may need to be adjusted through a series of colour changes and rotations.

Colour changes are straightforward: if a node is red, it is changed to black, and vice versa. Rotations, on the other hand, are a bit more complex. There are two types of rotations: left rotation and right rotation. A rotation operation preserves the in-order traversal property of the binary search tree, meaning that it maintains the order of the elements.

During an insertion or deletion, the tree is traversed from the root to the appropriate leaf, making rotations and changing colours along the way as necessary. After the operation, a final check is made to ensure the root is black.

In summary, a red-black tree is balanced by enforcing a set of properties through colour changes and rotations during insertions and deletions. This ensures that the tree remains approximately balanced, resulting in efficient search, insertion, and deletion operations.