---
id: cs-tries-trees
title: Tries / Trees — Prefix / Autocomplete / Routing
category: Coding
status: draft
source_trust_level: B
verification_status: conceptual
created_at: 2026-05-09
updated_at: 2026-05-09
tags: [cs, tree, trie, vibe-coding]
tech_stack: { language: "TS", applicable_to: ["Backend", "Frontend"] }
applied_in: []
aliases: [Trie, prefix tree, radix tree, ART, autocomplete, route matching, suffix tree]
---

# Tries / Trees

> Prefix-based 자료구조. **Autocomplete, route match, IP routing, dictionary**. Trie / Radix / ART (Adaptive Radix Tree). String key 가 자연.

## 📖 핵심 개념
- Trie: 매 char 가 node.
- Radix: 같은 path 압축.
- ART: cache-friendly, modern.
- Suffix tree: 모든 suffix 의 trie.

## 💻 코드 패턴

### Basic Trie
```ts
class TrieNode {
  children = new Map<string, TrieNode>();
  isEnd = false;
}

class Trie {
  root = new TrieNode();
  
  insert(word: string) {
    let node = this.root;
    for (const ch of word) {
      if (!node.children.has(ch)) {
        node.children.set(ch, new TrieNode());
      }
      node = node.children.get(ch)!;
    }
    node.isEnd = true;
  }
  
  search(word: string): boolean {
    const node = this.findNode(word);
    return node?.isEnd ?? false;
  }
  
  startsWith(prefix: string): boolean {
    return this.findNode(prefix) !== null;
  }
  
  private findNode(s: string): TrieNode | null {
    let node = this.root;
    for (const ch of s) {
      const next = node.children.get(ch);
      if (!next) return null;
      node = next;
    }
    return node;
  }
}
```

### Autocomplete
```ts
class AutocompleteTrie {
  // ... 위 + 
  
  suggestions(prefix: string, max = 10): string[] {
    const node = this.findNode(prefix);
    if (!node) return [];
    
    const result: string[] = [];
    this.collect(node, prefix, result, max);
    return result;
  }
  
  private collect(node: TrieNode, current: string, result: string[], max: number) {
    if (result.length >= max) return;
    if (node.isEnd) result.push(current);
    
    for (const [ch, child] of node.children) {
      this.collect(child, current + ch, result, max);
    }
  }
}

const trie = new AutocompleteTrie();
['apple', 'app', 'application', 'apply'].forEach(w => trie.insert(w));
trie.suggestions('app');  // ['app', 'apple', 'application', 'apply']
```

### Frequency-based autocomplete
```ts
class FrequencyTrie {
  root = new TrieNode();
  
  insert(word: string, freq: number = 1) {
    let node = this.root;
    for (const ch of word) {
      if (!node.children.has(ch)) {
        node.children.set(ch, new TrieNode());
      }
      node = node.children.get(ch)!;
    }
    node.frequency = (node.frequency ?? 0) + freq;
    node.word = word;
  }
  
  topSuggestions(prefix: string, k = 5): string[] {
    const node = this.findNode(prefix);
    if (!node) return [];
    
    // Heap 또는 sort
    const all: { word: string; freq: number }[] = [];
    this.collectAll(node, all);
    
    return all
      .sort((a, b) => b.freq - a.freq)
      .slice(0, k)
      .map(x => x.word);
  }
}
```

→ Search query autocomplete.

### Radix tree (compressed trie)
```ts
// "apple", "app", "apply"
// Trie:  a→p→p→l→e (end), p (end), p→l→y (end)
// Radix: "app" (end) → "le" (end), "ly" (end)
//                          ↳ "ication" (end)

class RadixNode {
  children = new Map<string, RadixNode>();  // edge label → node
  isEnd = false;
  value?: any;
}

class RadixTree {
  root = new RadixNode();
  
  insert(key: string, value: any) {
    // Common prefix 찾기 → split or extend
    // ... 복잡 implementation
  }
}
```

→ Memory 절약. URL routing 자주.

### URL routing (radix tree)
```
GET /users/:id
GET /users/:id/posts
GET /posts/:id
POST /posts

Tree:
/
├── users/
│   └── :id/
│       └── posts/
└── posts/
    └── :id (또는 default)
```

```ts
// find-my-way (Fastify 사용)
import findMyWay from 'find-my-way';

const router = findMyWay();
router.on('GET', '/users/:id', (req, res, params) => {
  res.end(`User ${params.id}`);
});

const match = router.find('GET', '/users/123');
// { handler, params: { id: '123' } }
```

→ Express / Fastify / Hono 의 router internals.

### IP routing (longest prefix match)
```
192.168.1.0/24 → router A
192.168.0.0/16 → router B
0.0.0.0/0 → router C (default)

→ Trie of bits.
```

```ts
class IPTrie {
  // Each bit (0 / 1) = child
  // Leaf = next-hop
}
```

→ Linux kernel routing.

### Suffix tree
```
"banana" 의 모든 suffix:
- banana
- anana
- nana
- ana
- na
- a

Suffix tree = 이 suffix 모두 의 trie (compressed).
```

```ts
// Substring search 빠름 (O(m), m = pattern length).
// Build = O(n).
// Use case: bioinformatics, text search.
```

→ Ukkonen's algorithm.

### Aho-Corasick (multi-pattern)
```ts
// 여러 pattern 을 한 번에 search.
// Trie + failure link.

const ac = new AhoCorasick();
ac.add('cat');
ac.add('dog');
ac.add('cattle');
ac.build();

const matches = ac.search('thecattleshookhead');
// [{ pattern: 'cat', start: 3 }, { pattern: 'cattle', start: 3 }]
```

→ Spam filter, DNA search, IDS.

### Prefix sum (different from trie)
```ts
// "ABC" → counts at each position
const prefix: number[] = [0];
for (const ch of str) prefix.push(prefix[prefix.length - 1] + (ch === 'a' ? 1 : 0));

// Range query: prefix[r] - prefix[l]
```

### Segment tree
```ts
// Range query / range update.
// 매 node 가 range 의 sum / min / max.

class SegmentTree {
  tree: number[];
  n: number;
  
  constructor(arr: number[]) {
    this.n = arr.length;
    this.tree = new Array(4 * this.n);
    this.build(arr, 0, 0, this.n - 1);
  }
  
  query(l: number, r: number): number {
    return this.queryHelper(0, 0, this.n - 1, l, r);
  }
  
  update(idx: number, val: number) {
    this.updateHelper(0, 0, this.n - 1, idx, val);
  }
}
```

→ Range sum / max / min 자주.

### Fenwick tree (BIT)
```ts
// Range sum + point update.
// Segment tree 보다 작음.

class BIT {
  tree: number[];
  
  constructor(n: number) {
    this.tree = new Array(n + 1).fill(0);
  }
  
  update(i: number, delta: number) {
    for (; i < this.tree.length; i += i & -i) this.tree[i] += delta;
  }
  
  query(i: number): number {
    let sum = 0;
    for (; i > 0; i -= i & -i) sum += this.tree[i];
    return sum;
  }
}
```

→ Inversion count, range sum.

### Splay tree / Red-black tree / AVL
```
Self-balancing BST.
- Splay: recently used = root (cache friendly)
- Red-black: balance via color
- AVL: balance via height

Used in:
- TreeMap / TreeSet (Java)
- std::map (C++)
- Linux kernel (Red-black for processes)
```

### B-tree (DB index)
```
[[CS_BTree_LSM_Storage]]:

매 node 가 multiple key (10-100s).
Disk-friendly.
Postgres / MySQL InnoDB.
```

### Patricia trie (compressed binary)
```
Bits 의 radix tree.
- IP routing
- Bitcoin merkle patricia (Ethereum state)
```

### MerkleTrie (Ethereum)
```
Hash 가 children 의 hash:
- Tamper detection
- Light client (proof)
```

### k-d tree (k-dimensional)
```
N-dim points 의 BST.
Use:
- Nearest neighbor search
- Range query
- 2D / 3D point cloud
```

```ts
class KDTree {
  // Each node split by 1 dim.
  // Alternate dimensions.
}

// 또는 외부 lib
import { kdTree } from 'kd-tree-javascript';
const tree = new kdTree(points, distance, ['x', 'y', 'z']);
const nearest = tree.nearest({ x: 0, y: 0, z: 0 }, 5);  // top 5
```

### Quadtree (2D 공간)
```ts
// Game collision, geo search.
// 매 node = 4 quadrants.

class Quadtree {
  bounds: Rect;
  points: Point[];
  children: Quadtree[] = [];
  
  insert(p: Point) {
    if (this.children.length > 0) {
      const idx = this.getIdx(p);
      this.children[idx].insert(p);
    } else {
      this.points.push(p);
      if (this.points.length > MAX_POINTS) this.split();
    }
  }
}
```

### Geohash
```
Lat/lon → string prefix.
"u4pruyd" — 0.6m precision.

Prefix match = nearby:
"u4pru" matches all in 5km of 'u4pru' area.

→ Trie + geo.
```

```ts
import geohash from 'ngeohash';

const hash = geohash.encode(37.5, 127.0, 9);  // 9 char ≈ 4.8m
const decoded = geohash.decode(hash);  // {latitude, longitude}
const neighbors = geohash.neighbors(hash);
```

### Use cases summary
```
Trie:
- Autocomplete (search box)
- Spell check
- IP routing
- Dictionary (English words)
- 회사 jargon

Radix:
- URL router (Express, Fastify)
- Memory-efficient string key

ART:
- In-memory DB (Hekaton)
- Cache-friendly

Suffix tree:
- DNA / bioinformatics
- Substring search

B-tree:
- DB index (Postgres, MySQL)
- File system (ext4)

Segment tree / BIT:
- Range query
- Competitive programming

k-d tree / quadtree:
- Geo search
- Game collision
```

### Performance
```
Trie operations:
- Insert / search: O(L) — L = key length
- Memory: O(N × L) — N = key count

Radix:
- Same as Trie + 작은 메모리 (compression)

Hash map (alternative):
- O(1) — but no prefix
- Use trie when prefix matters
```

### Trie vs hash map
```
Trie:
+ Prefix query (autocomplete)
+ Sorted order
+ Lex traversal
- 큰 메모리 (per char)

Hash map:
+ O(1) lookup
+ 작은 메모리
- No prefix
```

### Production library
```
- find-my-way: Fastify router (radix)
- ART: Adaptive Radix Tree (C / Rust)
- 자체: TS 직접 구현 OK
```

### When NOT to use trie
```
- Prefix 안 필요 (Hash map)
- 큰 string + 적은 query (Bloom filter)
- Memory critical (hash + Bloom)
```

## 🤔 의사결정 기준
| 사용 | 추천 |
|---|---|
| Autocomplete | Trie / Radix |
| URL routing | Radix tree |
| IP routing | Patricia / Radix bit |
| Substring search 큰 | Suffix tree / Aho-Corasick |
| Range query | Segment / BIT |
| Geo search | Quadtree / k-d tree / Geohash |
| In-memory DB | ART |

## ❌ 안티패턴
- **모든 곳 Trie**: hash map 충분 자주.
- **Trie 의 메모리 무 측정**: 큰 dataset = OOM.
- **Recursion depth (deep trie)**: stack overflow. iterative.
- **String key 만 가정**: binary trie 도 가능.
- **Suffix tree O(n²) build**: O(n) Ukkonen's.

## 🤖 LLM 활용 힌트
- Autocomplete = Trie 의 자연 use case.
- URL router 안 Radix tree.
- Geo = Geohash + Quadtree.
- DB = B-tree (다른 문서).

## 🔗 관련 문서
- [[CS_BTree_LSM_Storage]]
- [[CS_Big_O_Practical]]
- [[DB_Full_Text_Search]]