A Segment Tree is a binary tree used for storing intervals or segments. It allows querying which segments or intervals contain a given point. Segment trees are useful in scenarios requiring frequent updates and queries on a set of intervals.

The core operations of a Segment Tree include building the tree from a given array, updating values in the tree to reflect changes in the array, and querying the tree to obtain data over a range.

Building a segment tree involves constructing a binary tree where each leaf represents an element of the array, and each node stores information (like sum, minimum, or maximum) about the interval or segment it covers. This process typically runs in O(n log n) time, setting up the structure for efficient future updates and queries.

The SegmentTree class constructs a segment tree from an array for efficient range queries and updates. First, it doubles the array size for tree construction, placing each array element in a leaf node. The tree is then built upward by summing child nodes into their parent, enabling fast operations on array segments in logarithmic time.

Querying a segment tree efficiently retrieves the sum of a range within an array. Simply put, traverse down the tree from the root to the relevant leaves that represent the query interval. Since the segment tree is a binary tree, this operation can retrieve sums in O(log n) time, making it significantly faster than a linear approach using a for loop.

The query method begins at the root and selectively navigates to child nodes that overlap with the desired query range. If a segment completely falls within the query range, its sum is included in the result. If not, the tree further subdivides into smaller segments, efficiently combining the results of overlapping nodes.

Update a segment tree by modifying an element of the array and reflecting this change throughout the tree. This ensures the segment tree maintains accurate collection of data after array elements change. To update, the tree modifies the specific leaf node representing the array element and then updates all parent nodes to reflect this change.

The update method in the SegmentTree class modifies a single element by adjusting the value at the corresponding leaf and then recursively recalculating the sum for each parent node up to the root. This ensures the tree always represents the correct sums across all segments.

Constructed from an array, a segment tree transforms each array element into a leaf node and organizes parent nodes to calculate values from their children. This structure allows for efficient data operations across intervals.

Query operations use the tree's structure to efficiently compute sums, while updates are seamlessly handled by adjusting an element in the array and propagating this change throughout the tree. This ensures that the data remains consistent and accurately represents the current state of the array.

Lazy propagation is a technique used in segment trees to defer updates until necessary, optimizing performance for range queries. Instead of immediately updating all affected nodes upon an update, lazy propagation postpones the update to a later time, only applying it when needed during a query.

The key idea behind lazy propagation is to store pending updates at each node of the segment tree. When a range update is performed, instead of immediately updating the nodes, the update is stored as a "lazy" value at the node. During future queries, these lazy updates are applied as needed.

Segment trees should be your go-to tool for solving range query problems efficiently in coding interviews. They allow rapid updates and queries, making them great for dynamic data where values change frequently. Here are some essential tips and tricks to master segment trees: