Define a segment tree and its applications.

A segment tree is a data structure used for storing information about segments or intervals and allows for efficient range queries.

A segment tree is a binary tree where each node represents an interval or segment of an array. The root of the tree represents the entire array, and each child node represents a half segment of the parent node. The leaves of the tree represent individual elements of the array. The tree is constructed in a way that allows for efficient querying and updating of ranges within the array.

The primary use of a segment tree is to perform range queries, which are queries that ask for some information about a range of elements in an array. For example, a range query might ask for the sum, minimum, or maximum of a range of elements. Without a segment tree, a range query would require scanning through all the elements in the range, resulting in a time complexity of O(n). However, with a segment tree, the time complexity of a range query can be reduced to O(log n).

Segment trees can also be used to perform range updates, which are operations that modify a range of elements in an array. For example, a range update might increment all the elements in a range by a certain value. Like range queries, range updates can be performed more efficiently with a segment tree than without.

In addition to range queries and updates, segment trees can also be used for other types of queries and updates that involve intervals or segments. For example, they can be used to find the next smaller or larger element in a range, to count the number of distinct elements in a range, or to find the most frequent element in a range.

In summary, a segment tree is a versatile and efficient data structure for dealing with intervals or segments. It is particularly useful in situations where there are many range queries or updates, and where the array is not modified frequently.