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Intermediate Topic 26 of 52 Data Structures

Fenwick Trees

Compact structure for prefix sums with point updates.

How it works

A Fenwick tree, also called a binary indexed tree, stores partial sums in a single array and uses the lowest set bit of an index, i & -i, to decide which range each cell covers and how to move between cells. Prefix-sum queries and point updates both take O(log n) time and only O(n) memory with very small constants.

Where it is used

Counting inversions, maintaining cumulative frequencies, order-statistic queries, rank computations in coordinate-compressed data and 2D variants for image sums use Fenwick trees. They are favored for their short code.

Common pitfall

Indexing is one-based, and using index zero loops forever. They directly support only invertible operations like sum, not min or max. Range updates need a pair of trees. Remember to initialize correctly with a linear-time build instead of n updates when n is large.

Where it fits in the course

Fenwick Trees is topic 26 of 52 in Data Structures, rated Intermediate. It follows Segment Trees. The next topic is Graphs.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Data Structures quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Fenwick Trees?
Compact structure for prefix sums with point updates. It belongs to Data Structures and is an intermediate-level topic.
Where do people use Fenwick Trees?
Counting inversions, maintaining cumulative frequencies, order-statistic queries, rank computations in coordinate-compressed data and 2D variants for image sums use Fenwick trees.
What is the most common mistake with Fenwick Trees?
Indexing is one-based, and using index zero loops forever.
What should I learn before this topic?
Start with Segment Trees, the topic just before it in Data Structures.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Data Structures quiz. You can also explore Binary Search Tree in the Visual Lab.
Is Fenwick Trees hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Fenwick Trees?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Fenwick Trees?
Segment Trees, Graphs, Tries are closely connected. The full list is in the related topics section of this page.
What should I study after Fenwick Trees?
Continue with Graphs, the next topic in Data Structures.

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