Fenwick trees (Binary Indexed Trees) in Data Structures Theory - Step-by-Step Execution

Concept Flow - Fenwick trees (Binary Indexed Trees)

Start with array

↓

Build Fenwick Tree

↓

Query prefix sum

↓

Update element

↓

Repeat queries/updates

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End

Fenwick tree starts by building from an array, then supports fast prefix sum queries and updates by moving through tree nodes using bit operations.

Execution Sample

Data Structures Theory

arr = [1,2,3,4,5]
fenw = [0]*6

def update(i, val):
    while i < len(fenw):
        fenw[i] += val
        i += i & (-i)

def query(i):
    s = 0
    while i > 0:
        s += fenw[i]
        i -= i & (-i)
    return s

# Build fenw
for i in range(1,6):
  update(i, arr[i-1])

# Query sum(1..3)
print(query(3))

Build Fenwick tree from array and query sum of first 3 elements.

Analysis Table

Step	i	Action	Fenw array state	Output
1	1	update(1,1)	[0,1,1,0,1,0]
2	2	update(2,2)	[0,1,3,0,3,0]
3	3	update(3,3)	[0,1,3,3,6,0]
4	4	update(4,4)	[0,1,3,3,10,0]
5	5	update(5,5)	[0,1,3,3,10,5]
6	3	query sum(3)	[0,1,3,3,10,5]	6
7	4	query sum(4)	[0,1,3,3,10,5]	10
8	5	query sum(5)	[0,1,3,3,10,5]	15
9	-	update(3,2) (add 2 to index 3)	[0,1,3,5,12,5]
10	5	query sum(5)	[0,1,3,5,12,5]	17
11	-	End	[0,1,3,5,12,5]

💡 Finished building Fenwick tree and performing queries and updates.

State Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	After 6	After 9	Final
fenw	[0,0,0,0,0,0]	[0,1,1,0,1,0]	[0,1,3,0,3,0]	[0,1,3,3,6,0]	[0,1,3,3,10,0]	[0,1,3,3,10,5]	[0,1,3,3,10,5]	[0,1,3,5,12,5]	[0,1,3,5,12,5]
output							6		17

Key Insights - 3 Insights

Why does the Fenwick tree array have one extra element compared to the input array?

Fenwick tree uses 1-based indexing internally for easier bit manipulation, so fenw array size is input size + 1 (see rows 1-5 in execution_table).

How does the update function affect multiple positions in the Fenwick tree?

Update adds the value to fenw[i] and then moves to next index by adding the last set bit of i, updating all relevant nodes (see steps 1-5 and 9).

Why does the query sum(3) return 6 in step 6?

Query sums fenw[3] + fenw[2] = 3 + 3 = 6, combining partial sums stored in Fenwick tree nodes (step 6 in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table at step 6. What is the prefix sum output for sum(3)?

A 9

B 5

C 6

D 3

💡 Hint

Check the Output column at step 6 in execution_table.

At which step does the Fenwick tree get updated to include an additional +2 at index 3?

A Step 5

B Step 9

C Step 3

D Step 7

💡 Hint

Look for update actions with value 2 in the Action column.

If the initial array had 6 elements instead of 5, how would the fenw array size change?

A Size would be 7

B Size would be 6

C Size would be 5

D Size would be 8

💡 Hint

Fenwick tree uses 1-based indexing, so fenw size = input size + 1.

Concept Snapshot

Fenwick trees store cumulative frequencies for fast prefix sums.
Use 1-based indexing array fenw.
Update adds value at index and moves up using i += i & (-i).
Query sums prefix by moving down using i -= i & (-i).
Both operations run in O(log n) time.

Full Transcript

Fenwick trees, also called Binary Indexed Trees, help quickly find sums of elements from the start of an array up to any position. We build a fenw array one element larger than the input to use 1-based indexing. To update, we add a value at an index and then move to the next relevant node by adding the last set bit of the index. To query a prefix sum, we add fenw values moving down by subtracting the last set bit until we reach zero. This process is efficient and runs in logarithmic time. The execution table shows building fenw from an array, querying sums, and updating values step-by-step.