Why is tree balancing necessary? How is fast search achieved?

Balancing a tree is necessary to maintain its structure so that the height of the tree remains as minimal as possible. This is critical for efficient execution of search, insert, and delete operations.

If the tree is unbalanced, it can become a structure close to a list, and operations will run in linear time O(n).

The quick search is achieved through:

• Balanced tree height: the height of a balanced tree grows proportionally to the logarithm of the number of elements (O(log n)).
• Fast level traversal: during search, we compare the key with the current node and move either to the left or right subtree, effectively discarding half of the elements.

Examples of balancing:

• AVL trees: maintain balance through rotations after insertion or deletion.
• Red-black trees: ensure balance using node color properties.

Thus, balancing guarantees that search, insert, and delete operations are performed in O(log n), which is significantly faster than in unbalanced structures.