---
title: "B-tree properties; order 5 for 3, 14, 7, ..."
subject: "Data Structure and Algorithm"
chapter: 6
key: "semester_4/data_structure_and_algorithm:btree-properties-order5-3-14"
source: https://www.bctengineer.com/q/semester_4/data_structure_and_algorithm/btree-properties-order5-3-14
site: BCT Engineering
---

# B-tree properties; order 5 for 3, 14, 7, ...

Data Structure and Algorithm · Chapter 6: Trees

- Asked in: 2070 Bhadra, 2+3 marks
- Video: [5.25 Insertion in B-Tree of Order 5 | Data structures and algorithms Tutorials](https://www.youtube.com/watch?v=c7hXEFs69Jw) — 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-properties-order5-3-14
- Whole chapter: https://www.bctengineer.com/md/semester_4/data_structure_and_algorithm/chapter-6

## Question

Write down the properties of B-tree. Construct a B-tree of order 5 for following data: 3, 14, 7, 1, 8, 5, 11, 17, and 13.

## Short answer

B-tree properties: max m children, min ⌈m/2⌉ children for non-root, all leaves same level, keys sorted, tree balanced. Constructed order-5 B-tree: root [7 13] with leaves [1 3 5] [8 11] [14 17].

## Answer

#### Properties of a B-tree of order m
1. Every node has at most m children, i.e. at most m − 1 keys.
2. Every non-root, non-leaf node has at least ⌈m/2⌉ children (and every non-root node has at least ⌈m/2⌉ − 1 keys).
3. The root has at least 2 children if it is not a leaf.
4. A node with k children holds k − 1 keys, stored in ascending order.
5. All leaves are on the same level, so the tree is perfectly height-balanced.
6. For a node with keys k₁ < k₂ < ..., the subtree to the left of k₁ holds keys smaller than k₁, the subtree between kᵢ and kᵢ₊₁ holds keys that lie between them, and the subtree to the right of the last key holds larger keys.
7. The tree grows and shrinks at the root; search, insert and delete take O(log n).

#### B-tree of order 5 (max 4 keys, min 2 keys per non-root node)

| Step | Insert | Tree after step | Action |
|---|---|---|---|
| 1 | 3 | `[3]` | fits in leaf |
| 2 | 14 | `[3 14]` | fits in leaf |
| 3 | 7 | `[3 7 14]` | fits in leaf |
| 4 | 1 | `[1 3 7 14]` | fits in leaf |
| 5 | 8 | `[7]{[1 3], [8 14]}` | leaf [1 3 7 8 14] overflows, split: 7 moves up; new root |
| 6 | 5 | `[7]{[1 3 5], [8 14]}` | fits in leaf |
| 7 | 11 | `[7]{[1 3 5], [8 11 14]}` | fits in leaf |
| 8 | 17 | `[7]{[1 3 5], [8 11 14 17]}` | fits in leaf |
| 9 | 13 | `[7 13]{[1 3 5], [8 11], [14 17]}` | leaf [8 11 13 14 17] overflows, split: 13 moves up |

#### Final tree

```text
[7 13]
[1 3 5]  [8 11]  [14 17]
```

## Check yourself

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

1. In a B-tree of order m, how many keys can a node hold at most?
   A. m - 1
   B. m
   C. m + 1
   D. 2m
   Answer: A. Each node has at most m children and so at most m - 1 keys. (Key point: Key limits)

2. What is true of the leaves of a B-tree?
   A. Leaves may be at any level
   B. Only the root can be a leaf
   C. All leaves are on the same level
   D. Leaves hold no keys
   Answer: C. This makes the tree perfectly height-balanced. (Key point: Leaf level)

3. In order m, a non-root internal node has at least how many children?
   A. ceil(m/2)
   B. 2
   C. m - 1
   D. m
   Answer: A. For m = 5 this is 3 children (2 keys). (Key point: Non-root minimum)

4. When 13 is inserted into [8 11 14 17] (order 5), what happens?
   A. Split with 11 moving up
   B. The root becomes [13]
   C. Overflow: split at median 13, which moves up to join 7 in the root
   D. Nothing; the leaf holds 5 keys
   Answer: C. [8 11 13 14 17] has median 13, giving root [7 13] and leaves [1 3 5], [8 11], [14 17]. (Key point: Construction split)

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