---
title: "B-tree of order 3 for 52, 46, 27, ..."
subject: "Data Structure and Algorithm"
chapter: 6
key: "semester_4/data_structure_and_algorithm:btree-order3-52-46-27"
source: https://www.bctengineer.com/q/semester_4/data_structure_and_algorithm/btree-order3-52-46-27
site: BCT Engineering
---

# B-tree of order 3 for 52, 46, 27, ...

Data Structure and Algorithm · Chapter 6: Trees

- Asked in: 2072 Magh, 6 marks
- Video: [5.24 Insertion in B-tree of Order 3 | B-Tree Example | Data structures and algorithms](https://www.youtube.com/watch?v=Ay2AbTk_QEg) — Jenny's Lectures CS IT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/data_structure_and_algorithm/btree-order3-52-46-27
- Whole chapter: https://www.bctengineer.com/md/semester_4/data_structure_and_algorithm/chapter-6

## Question

Construct a B-tree of order 3 for given set of data: 52, 46, 27, 81, 90, 108, 72, 110, 35, 115, 121 and 86.

## Short answer

Insert sequentially; node overflows at 3 keys, splits. Final B-tree order 3: root [81], left internal [46] with leaves [27 35] [52 72]; right internal [108 115] with leaves [86 90] [110] [121].

## Answer

A B-tree of order 3 allows at most 3 children (2 keys) per node and at least 1 key in a non-root node. A key is inserted into its leaf; if the leaf then holds 3 keys it overflows, is split at the median, and the median moves up to the parent. The split may repeat up to the root.

#### Step-by-step construction

| Step | Insert | Tree after step | Action |
|---|---|---|---|
| 1 | 52 | `[52]` | fits in leaf |
| 2 | 46 | `[46 52]` | fits in leaf |
| 3 | 27 | `[46]{[27], [52]}` | leaf [27 46 52] overflows, split: 46 moves up; new root |
| 4 | 81 | `[46]{[27], [52 81]}` | fits in leaf |
| 5 | 90 | `[46 81]{[27], [52], [90]}` | leaf [52 81 90] overflows, split: 81 moves up |
| 6 | 108 | `[46 81]{[27], [52], [90 108]}` | fits in leaf |
| 7 | 72 | `[46 81]{[27], [52 72], [90 108]}` | fits in leaf |
| 8 | 110 | `[81]{[46]{[27], [52 72]}, [108]{[90], [110]}}` | leaf [90 108 110] overflows, split: 108 moves up; internal node [46 81 108] overflows, split: 81 moves up; new root |
| 9 | 35 | `[81]{[46]{[27 35], [52 72]}, [108]{[90], [110]}}` | fits in leaf |
| 10 | 115 | `[81]{[46]{[27 35], [52 72]}, [108]{[90], [110 115]}}` | fits in leaf |
| 11 | 121 | `[81]{[46]{[27 35], [52 72]}, [108 115]{[90], [110], [121]}}` | leaf [110 115 121] overflows, split: 115 moves up |
| 12 | 86 | `[81]{[46]{[27 35], [52 72]}, [108 115]{[86 90], [110], [121]}}` | fits in leaf |

#### Final B-tree of order 3

```text
[81]
[46]  [108 115]
[27 35]  [52 72]  [86 90]  [110]  [121]
```
All leaves are on the same level and every node respects the key limits.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In a B-tree of order 3, a node overflows when it has how many keys?
   A. 2 keys
   B. 3 keys
   C. 4 keys
   D. 1 key
   Answer: B. Order 3 allows at most 2 keys; a third key triggers a split. (Key point: Order 3 limits)

2. After 52, 46 and 27, which key becomes the root?
   A. 27
   B. 52
   C. The root remains [27 46 52]
   D. 46
   Answer: D. The median of 27 46 52 is 46, which moves up to a new root. (Key point: First split)

3. Inserting 110 leads to what?
   A. Only a leaf split
   B. Leaf split (108 up) and then internal node split (81 up), giving a new root [81]
   C. Rotation like an AVL tree
   D. 110 becomes the root
   Answer: B. [90 108 110] splits, pushing 108 to [46 81] making it [46 81 108], which overflows and pushes 81 up. (Key point: Cascading split with 110)

4. How many leaves does the final tree have and where are they?
   A. 3 leaves at different levels
   B. 4 leaves
   C. 7 leaves
   D. 5 leaves, all at the same level
   Answer: D. The final leaves are [27 35], [52 72], [86 90], [110], [121], all at the same depth. (Key point: Final shape)

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