---
id: wiki-2026-0508-b-tree
title: B-Tree
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [B+Tree, BTree, Balanced-Tree-Index]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [data-structure, tree, index, database]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: stdlib
---

# B-Tree

## 매 한 줄
> **"매 disk-friendly한 self-balancing search tree — 매 한 노드에 매 많은 key를 저장해 매 height를 minimize"**. 매 1970년 Bayer & McCreight가 IBM에서 design, 매 2026 PostgreSQL/MySQL InnoDB/SQLite의 default index, 매 NVMe SSD에서도 여전히 dominant — 매 sequential I/O와 cache line alignment 친화적.

## 매 핵심

### 매 invariant
- 매 node는 매 $[t-1, 2t-1]$ keys 보유 (매 root만 예외).
- 매 internal node는 매 $[t, 2t]$ children.
- 매 모든 leaf는 매 same depth.
- 매 keys 매 sorted within node.

### 매 B+ Tree variant (DB 표준)
- 매 internal node는 매 keys만 — 매 data는 매 leaf에만.
- 매 leaf끼리 매 linked list — 매 range scan $O(k)$.
- 매 PostgreSQL/MySQL이 매 사용.

### 매 응용
1. RDBMS index (PostgreSQL btree).
2. Filesystem (ext4 HTree, NTFS).
3. KV store (LevelDB SST, RocksDB).
4. Vector DB metadata index.

## 💻 패턴

### B-Tree node (Python)
```python
class BTreeNode:
    def __init__(self, t, leaf=False):
        self.t = t              # min degree
        self.keys = []
        self.children = []
        self.leaf = leaf

    def search(self, k):
        i = 0
        while i < len(self.keys) and k > self.keys[i]:
            i += 1
        if i < len(self.keys) and self.keys[i] == k:
            return (self, i)
        if self.leaf:
            return None
        return self.children[i].search(k)
```

### Insert with split
```python
def split_child(parent, i):
    t = parent.t
    full = parent.children[i]
    new = BTreeNode(t, full.leaf)
    new.keys = full.keys[t:]
    if not full.leaf:
        new.children = full.children[t:]
        full.children = full.children[:t]
    parent.keys.insert(i, full.keys[t-1])
    full.keys = full.keys[:t-1]
    parent.children.insert(i+1, new)

def insert(root, k):
    if len(root.keys) == 2*root.t - 1:
        new_root = BTreeNode(root.t)
        new_root.children.append(root)
        split_child(new_root, 0)
        root = new_root
    insert_nonfull(root, k)
    return root
```

### B+ Tree range scan
```python
def range_scan(leaf, lo, hi):
    out = []
    node = leaf
    while node:
        for k in node.keys:
            if lo <= k <= hi: out.append(k)
            elif k > hi: return out
        node = node.next  # leaf-linked list
    return out
```

### PostgreSQL B-Tree usage
```sql
CREATE INDEX idx_users_email ON users USING btree (email);
-- equality + range + sort 사용
EXPLAIN SELECT * FROM users WHERE email > 'a' AND email < 'm';
-- Index Scan using idx_users_email
```

### SQLite WITHOUT ROWID (B-Tree direct)
```sql
CREATE TABLE kv (k TEXT PRIMARY KEY, v BLOB) WITHOUT ROWID;
-- 매 data가 매 PK index 자체에 — 매 secondary lookup 제거
```

### Bulk loading (sorted insert)
```python
def bulk_load(sorted_pairs, t=64):
    # Sort + bottom-up build (vs O(n log n) per-insert)
    leaves = [sorted_pairs[i:i+2*t-1]
              for i in range(0, len(sorted_pairs), 2*t-1)]
    # build internal levels...
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| OLTP, point + range query | B+ Tree (default) |
| Append-heavy timeseries | LSM (RocksDB) — B-Tree write amp 高 |
| In-memory only, no range | Hash index |
| Vector similarity | HNSW (not B-Tree) |
| Spatial | R-Tree / GiST |

**기본값**: 매 RDBMS index는 매 B+ Tree. 매 SSD/NVMe에서도 매 page-aligned (8KB-16KB) 노드.

## 🔗 Graph
- 부모: [[Linked-Lists-and-Trees]] · [[Theoretical-Computer-Science]]
- 변형: [[Hash-Functions-and-Maps]] (alternative)
- 응용: Database-Index · [[Bloom-Filters in Search]]
- Adjacent: [[Algorithm-Complexity-Big-O]] · LSM-Tree

## 🤖 LLM 활용
**언제**: 매 DB schema design, 매 "왜 query가 slow?" debugging, 매 index choice review.
**언제 X**: 매 in-memory + write-heavy → LSM 우선; 매 vector search → HNSW.

## ❌ 안티패턴
- **Random UUID v4 PK**: 매 B-Tree에 매 random insert → 매 page split storm. 매 UUIDv7 (time-ordered) 사용.
- **Over-indexing**: 매 모든 column에 index — 매 write amp + storage 폭증.
- **Index on low-cardinality**: 매 boolean column index — 매 useless, full scan 더 빠름.

## 🧪 검증 / 중복
- Verified (Bayer 1972 original paper, PostgreSQL docs Ch.62).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — B-Tree/B+Tree, split logic, DB index |