B+ trees in databases in Data Structures Theory - Step-by-Step Execution

Concept Flow - B+ trees in databases

Start at Root Node

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Search Key in Node

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Is Node Leaf?

No→Follow Child Pointer to Next Node

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Repeat Search in Child Node

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Find Key or Position

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If Insert: Add Key

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Is Node Full?

No→Done

Yes↓

Split Node

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Push Up Middle Key to Parent

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If Parent Full, Repeat Split

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Done

This flow shows how a B+ tree searches for a key starting at the root, moves down internal nodes, and inserts keys by splitting nodes when full, pushing keys up to parents.

Execution Sample

Data Structures Theory

Insert keys: 10, 20, 5, 6, 12 into B+ tree of order 3

Steps:
1. Insert 10
2. Insert 20
3. Insert 5
4. Insert 6 (causes split)
5. Insert 12

This example shows inserting keys into a B+ tree with max 3 keys per node, demonstrating node splits and key promotion.

Analysis Table

Step	Action	Node State Before	Operation	Node State After	Notes
1	Insert 10	Empty root	Add 10 to root	Root keys: [10]	Root is leaf, no split
2	Insert 20	Root keys: [10]	Add 20 to root	Root keys: [10, 20]	Still space, no split
3	Insert 5	Root keys: [10, 20]	Add 5 and sort	Root keys: [5, 10, 20]	Node full (3 keys)
4	Insert 6	Root keys: [5, 10, 20]	Add 6 and sort: [5, 6, 10, 20]	Split node into two: Left: [5, 6] Right: [10, 20]	Push up 10 to new root; height increases to 2
5	Insert 12	Root keys: [10] Right child: [10, 20]	Add 12 to right child and sort	Right child keys: [10, 12, 20]	Right child full
6	Split right child	Right child keys: [10, 12, 20]	Split into [10] and [12, 20]	Root keys: [10, 12] Children: Left [5, 6], Middle [10], Right [12, 20]	Push up 12 to root; root has space, no further split
7	Done	Final tree	No operation	Root keys: [10, 12] Leaves: [5, 6], [10], [12, 20]	Insertion complete

💡 All keys inserted; tree balanced with splits and key promotions.

State Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	Final
Root keys	[]	[10]	[10, 20]	[5, 10, 20]	[10]	[10]	[10, 12]	[10, 12]
Left child keys	N/A	N/A	N/A	N/A	[5, 6]	[5, 6]	[5, 6]	[5, 6]
Right child keys	N/A	N/A	N/A	N/A	[10, 20]	[10, 12, 20]	[12, 20]	[12, 20]
Tree height	1	1	1	1	2	2	2	2

Key Insights - 3 Insights

Why do we split the node when it becomes full?

When a node has more keys than allowed (step 4 in execution_table), it splits to keep the tree balanced and maintain search efficiency.

What happens to the middle key during a split?

The middle key is pushed up to the parent node (step 4), acting as a separator between the two new child nodes.

Why does the tree height increase after some splits?

When the root node is full and splits (step 4), a new root is created, increasing the tree height by one.

Visual Quiz - 3 Questions

Look at the execution_table at step 4. What keys are in the left child after the split?

A [5, 10]

B [5, 6]

C [10, 20]

D [6, 10]

💡 Hint

Check the 'Node State After' column in step 4 for the split details.

At which step does the tree height increase?

A Step 6

B Step 3

C Step 4

D Step 2

💡 Hint

Look at the 'Tree height' variable in variable_tracker after each step.

If we insert a key smaller than all existing keys at step 5, how would the execution_table change?

A The left child would receive the new key instead

B The right child would split earlier

C The root would split immediately

D No splits would occur

💡 Hint

Consider where keys smaller than existing ones go in a B+ tree (see step 5 node states).

Concept Snapshot

B+ trees store sorted keys in nodes with pointers to children.
Leaf nodes hold actual data keys.
When a node is full, it splits and pushes the middle key up.
This keeps the tree balanced and search fast.
Tree height grows only when root splits.
Used in databases for efficient range queries.

Full Transcript

This visual execution trace shows how B+ trees in databases manage keys and maintain balance. Starting from an empty root, keys are inserted one by one. When a node reaches its maximum capacity, it splits into two nodes, and the middle key is pushed up to the parent node. If the parent is also full, it splits too, possibly increasing the tree height. Leaf nodes contain the actual data keys, and internal nodes guide the search. This structure ensures fast search, insert, and range queries by keeping the tree balanced. The example inserts keys 10, 20, 5, 6, and 12 into a B+ tree of order 3, showing node splits and key promotions step-by-step.