Jul 22, 2026knoxiboy

Trie (Prefix Tree)

A Trie (pronounced "try", from retrieval) is a tree-based data structure designed for storing and searching strings with shared prefixes. Unlike a HashMap, a Trie lets you search by prefix in O(m) time where m is the length of the word — making it the backbone of search autocomplete, spell checkers, and IP routing.

Video Explanation

Data Structures: Tries - YouTube thumbnail

🧠 Why Trie? — Comparison with Other Structures

Store words: ["apple", "app", "apt", "bat", "ball"]
Then answer: "Give all words starting with 'ap'"

❌ Array/List:
  Scan every word → O(n × m) per query → slow for large dictionaries

❌ HashMap:
  O(1) exact lookup but NO prefix search support
  "app" → exists, but "give all words starting with ap" requires scanning all keys

✅ Trie:
  Insert: O(m) per word
  Prefix search: O(m) to reach prefix node, then collect all words below → Fast!

🌳 Trie Structure — Visual Diagram

Words inserted: ["app", "apple", "apt", "bat", "ball"]

                root
               /    \
              a      b
              |      |
              p      a
             / \    / \
            p   t  t   l
            |   |  |   |
           [✓] [✓][✓]  l
            |           |
            l           [✓]
            |
            e
            |
           [✓]

[✓] = isEndOfWord is true

Paths from root:
root→a→p→p        = "app"   ✓
root→a→p→p→l→e    = "apple" ✓
root→a→p→t        = "apt"   ✓
root→b→a→t        = "bat"   ✓
root→b→a→l→l      = "ball"  ✓

🔩 Node Structure

Each Trie node contains:
1. children[26]  → array of 26 pointers (one per lowercase letter a-z)
                   children[0] = pointer to 'a' child
                   children[1] = pointer to 'b' child
                   ...
                   children[25] = pointer to 'z' child
2. isEndOfWord   → boolean flag: true if a valid word ends at this node

Initially all children = null, isEndOfWord = false

➕ Operation 1: Insert

Visual Dry-Run — Insert "app"

Start at root. Word = "app"

Character 'a':
  root.children[0] == null? → YES → create new node
  Move to children[0]

Character 'p':
  node.children[15] == null? → YES → create new node
  Move to children[15]

Character 'p':
  node.children[15] == null? → YES → create new node
  Move to children[15]

End of word → set isEndOfWord = true ✓

root → [a] → [p] → [p*]     (* = end of word)

Visual Dry-Run — Insert "apple" after "app"

Word = "apple"

'a' → node exists (root.children[0]) → just move, no new node
'p' → node exists → just move
'p' → node exists → just move  (this was end of "app", stays end)
'l' → null → create new node, move
'e' → null → create new node, move
End → isEndOfWord = true ✓

root → [a] → [p] → [p*] → [l] → [e*]

Code

Python
Java
C++

class TrieNode:
    def __init__(self):
        self.children = {}        # char → TrieNode
        self.is_end_of_word = False

class Trie:
    def __init__(self):
        self.root = TrieNode()

    def insert(self, word: str) -> None:
        node = self.root
        for char in word:
            if char not in node.children:
                node.children[char] = TrieNode()   # create node if missing
            node = node.children[char]              # move to next node
        node.is_end_of_word = True                  # mark end of word

# Example
trie = Trie()
trie.insert("app")
trie.insert("apple")
trie.insert("apt")

import java.util.HashMap;

class TrieNode {
    HashMap<Character, TrieNode> children;
    boolean isEndOfWord;

    public TrieNode() {
        children = new HashMap<>();
        isEndOfWord = false;
    }
}

public class Trie {
    protected TrieNode root;

    public Trie() {
        root = new TrieNode();
    }

    public void insert(String word) {
        TrieNode node = root;
        for (char c : word.toCharArray()) {
            if (!node.children.containsKey(c)) {
                node.children.put(c, new TrieNode());  // create node if missing
            }
            node = node.children.get(c);               // move to next node
        }
        node.isEndOfWord = true;                       // mark end of word
    }
}

#include <vector>
#include <string>
#include <iostream>
using namespace std;

struct TrieNode {
    vector<TrieNode*> children;
    bool isEndOfWord;

    TrieNode() : children(26, nullptr), isEndOfWord(false) {}
    ~TrieNode() {
        for (auto child : children) {
            delete child;
        }
    }
};

class Trie {
public:
    TrieNode* root;

    Trie() {
        root = new TrieNode();
    }

    ~Trie() {
        delete root;
    }

    void insert(const string& word) {
        TrieNode* node = root;
        for (char c : word) {
            int idx = c - 'a';
            if (!node->children[idx]) {
                node->children[idx] = new TrieNode();  // create node if missing
            }
            node = node->children[idx];                // move to next node
        }
        node->isEndOfWord = true;                      // mark end of word
    }
};

Time Complexity: O(m) where m = length of word Space Complexity: O(m) per word inserted

🔍 Operation 2: Search (Exact Word)

Visual Dry-Run

Trie contains: ["app", "apple", "apt"]

root → [a] → [p] → [p*] → [l] → [e*]
                  ↘ [t*]

Search "app":
'a' → exists → move
'p' → exists → move
'p' → exists → move
End → isEndOfWord = TRUE ✅ → "app" EXISTS

Search "ap":
'a' → exists → move
'p' → exists → move
End → isEndOfWord = FALSE ❌ → "ap" is NOT a word
(node exists as prefix but not marked as complete word)

Code

Python
Java
C++

    def search(self, word: str) -> bool:
        node = self.root
        for char in word:
            if char not in node.children:
                return False        # path doesn't exist
            node = node.children[char]
        return node.is_end_of_word  # true only if complete word

    public boolean search(String word) {
        TrieNode node = root;
        for (char c : word.toCharArray()) {
            if (!node.children.containsKey(c)) {
                return false;           // path doesn't exist
            }
            node = node.children.get(c);
        }
        return node.isEndOfWord;        // true only if complete word
    }

    bool search(const string& word) {
        TrieNode* node = root;
        for (char c : word) {
            int idx = c - 'a';
            if (!node->children[idx]) {
                return false;           // path doesn't exist
            }
            node = node->children[idx];
        }
        return node->isEndOfWord;       // true only if complete word
    }

Time Complexity: O(m) | Space Complexity: O(1)

🔎 Operation 3: StartsWith (Prefix Search)

Visual Dry-Run

startsWith("ap"):
'a' → exists → move
'p' → exists → move
Reached end of prefix → node exists → TRUE ✅

startsWith("bx"):
'b' → exists → move
'x' → does NOT exist → FALSE ❌

Key difference:
search("ap")     → FALSE (not a complete word)
startsWith("ap") → TRUE  (it IS a valid prefix)

Code

Python
Java
C++

    def starts_with(self, prefix: str) -> bool:
        node = self.root
        for char in prefix:
            if char not in node.children:
                return False        # prefix path doesn't exist
            node = node.children[char]
        return True                 # prefix exists

    public boolean startsWith(String prefix) {
        TrieNode node = root;
        for (char c : prefix.toCharArray()) {
            if (!node.children.containsKey(c)) {
                return false;           // prefix path doesn't exist
            }
            node = node.children.get(c);
        }
        return true;                    // prefix exists
    }

    bool startsWith(const string& prefix) {
        TrieNode* node = root;
        for (char c : prefix) {
            int idx = c - 'a';
            if (!node->children[idx]) {
                return false;           // prefix path doesn't exist
            }
            node = node->children[idx];
        }
        return true;                    // prefix exists
    }

Time Complexity: O(m) | Space Complexity: O(1)

🗑️ Operation 4: Delete

Visual Dry-Run

Delete "app" from Trie containing ["app", "apple", "apt"]

Before:
root → [a] → [p] → [p*] → [l] → [e*]
                  ↘ [t*]

Strategy: Only unmark isEndOfWord at the last node.
Do NOT delete the node — "apple" still needs this path!

After deleting "app":
root → [a] → [p] → [p] → [l] → [e*]   ← 'p' no longer end of word
                  ↘ [t*]

search("app")   → FALSE ✅ (deleted)
search("apple") → TRUE  ✅ (intact)

Code

Python
Java
C++

    def delete(self, word: str) -> bool:
        if not self.search(word):
            return False

        def _delete(node, word, depth):
            if depth == len(word):
                node.is_end_of_word = False
                return len(node.children) == 0

            char = word[depth]
            if _delete(node.children[char], word, depth + 1):
                del node.children[char]
                return len(node.children) == 0 and not node.is_end_of_word
            return False

        _delete(self.root, word, 0)
        return True

    public boolean delete(String word) {
        if (!search(word)) return false;
        deleteHelper(root, word, 0);
        return true;
    }

    private boolean deleteHelper(TrieNode node, String word, int depth) {
        if (depth == word.length()) {
            node.isEndOfWord = false;
            return node.children.isEmpty();
        }

        char c = word.charAt(depth);
        boolean shouldDelete = deleteHelper(node.children.get(c), word, depth + 1);

        if (shouldDelete) {
            node.children.remove(c);
            return node.children.isEmpty() && !node.isEndOfWord;
        }

        return false;
    }

    bool deleteWord(const string& word) {
        if (!search(word)) return false;
        deleteWordHelper(root, word, 0);
        return true;
    }

private:
    bool deleteWordHelper(TrieNode* node, const string& word, int depth) {
        if (depth == (int)word.size()) {
            node->isEndOfWord = false;
            for (auto child : node->children)
                if (child) return false;
            return true;
        }

        int idx = word[depth] - 'a';
        bool shouldDelete = deleteWordHelper(node->children[idx], word, depth + 1);

        if (shouldDelete) {
            delete node->children[idx];
            node->children[idx] = nullptr;
            if (!node->isEndOfWord) {
                for (auto child : node->children)
                    if (child) return false;
                return true;
            }
        }

        return false;
    }

Time Complexity: O(m) | Space Complexity: O(m) recursion stack

✨ Bonus: AutoComplete Feature

Visual Dry-Run

Trie: ["app", "apple", "apt", "bat", "ball"]
AutoComplete("ap"):

Step 1: Navigate to node for prefix "ap"
  root → 'a' → 'p' → [at this node]

Step 2: DFS from this node collecting all complete words
  → 'p'* → found "app"
       → 'l' → 'e'* → found "apple"
  → 't'* → found "apt"

Result: ["app", "apple", "apt"] ✅

Code

Python
Java
C++

class TrieWithAutoComplete(Trie):
    def autocomplete(self, prefix: str) -> list:
        node = self.root

        # Navigate to prefix node
        for char in prefix:
            if char not in node.children:
                return []
            node = node.children[char]

        # DFS to collect all words
        results = []
        self._dfs(node, prefix, results)
        return results

    def _dfs(self, node, current_word, results):
        if node.is_end_of_word:
            results.append(current_word)
        for char, child_node in node.children.items():
            self._dfs(child_node, current_word + char, results)

# Example
trie = TrieWithAutoComplete()
for word in ["app", "apple", "apt", "bat", "ball"]:
    trie.insert(word)

print(trie.autocomplete("ap"))   # ['app', 'apple', 'apt']
print(trie.autocomplete("ba"))   # ['bat', 'ball']
print(trie.autocomplete("xyz"))  # []

import java.util.ArrayList;
import java.util.List;

public class TrieWithAutoComplete extends Trie {

    public List<String> autocomplete(String prefix) {
        TrieNode node = root;
        List<String> results = new ArrayList<>();

        // Navigate to prefix node
        for (char c : prefix.toCharArray()) {
            if (!node.children.containsKey(c)) return results;
            node = node.children.get(c);
        }

        // DFS to collect all words
        dfs(node, new StringBuilder(prefix), results);
        return results;
    }

    private void dfs(TrieNode node, StringBuilder current, List<String> results) {
        if (node.isEndOfWord) {
            results.add(current.toString());
        }
        for (char c : node.children.keySet()) {
            current.append(c);
            dfs(node.children.get(c), current, results);
            current.deleteCharAt(current.length() - 1);  // backtrack
        }
    }
}

class TrieWithAutoComplete : public Trie {
public:
    vector<string> autocomplete(const string& prefix) {
        TrieNode* node = root;
        vector<string> results;

        // Navigate to prefix node
        for (char c : prefix) {
            int idx = c - 'a';
            if (!node->children[idx]) return results;
            node = node->children[idx];
        }

        // DFS to collect all words
        string current = prefix;
        dfs(node, current, results);
        return results;
    }

private:
    void dfs(TrieNode* node, string& current, vector<string>& results) {
        if (node->isEndOfWord) results.push_back(current);
        for (int i = 0; i < 26; i++) {
            if (node->children[i]) {
                current.push_back('a' + i);
                dfs(node->children[i], current, results);
                current.pop_back();     // backtrack
            }
        }
    }
};

⚡ Brute Force vs Trie

Dictionary: 100,000 words, average length 8
Query: "Find all words starting with 'pre'"

❌ BRUTE FORCE — O(n × m):
  Check every word: 100,000 × 8 = 800,000 operations per query

✅ TRIE — O(m + k):
  Navigate prefix "pre": 3 steps
  Collect k=500 matching words: 503 operations total

Speedup: ~1,600x faster ✅

📊 Complexity Summary

Operation	Time	Space
Insert	O(m)	O(m)
Search	O(m)	O(1)
StartsWith	O(m)	O(1)
Delete	O(m)	O(m) stack
AutoComplete	O(m + k)	O(k)

m = length of word/prefix | k = number of results returned

❌ Common Mistakes

Confusing search() and startsWith() — search("app") checks for a complete word. startsWith("app") checks for any word with that prefix. They return different results!
Deleting shared prefix nodes — Never delete a node if other words share its path. Always check children before removing.
Case sensitivity — Tries are case-sensitive. Always normalize input to lowercase before inserting or searching.
Array vs HashMap for children — Array children[26] is O(1) access but wastes memory. HashMap uses less memory but has overhead. Choose based on your character set size.
Forgetting isEndOfWord — Inserting "apple" without marking isEndOfWord=true causes search("apple") to return false even though the path exists.

🏋️ Practice Problems

#	Problem	Concept	Difficulty
1	Implement Trie (Prefix Tree)	Insert, Search, StartsWith	🟡 Medium
2	Longest Common Prefix	Trie traversal	🟢 Easy
3	Design Add and Search Words	Wildcard search in Trie	🟡 Medium
4	Replace Words	Prefix replacement	🟡 Medium
5	Map Sum Pairs	Trie with values	🟡 Medium
6	Word Search II	Trie + DFS on grid	🔴 Hard
7	Maximum XOR of Two Numbers	Binary Trie	🔴 Hard
8	Stream of Characters	Trie + sliding window	🔴 Hard

🌍 Real-World Applications

Search autocomplete — Google and Bing suggest completions as you type
Spell checker — Word processors verify if a word exists in the dictionary
IP routing (LPM) — Routers use binary Tries for Longest Prefix Match
T9 predictive text — Mobile keyboard word prediction
DNA sequence matching — Biological sequence databases

Segment Tree: Tree structure specialized for range queries and updates ( $O(\log N)$ ).
Disjoint Set Union (DSU): Manages dynamic connectivity and equivalence relations ( $O(\alpha(N))$ ).
Two Pointers & Sliding Window: Efficient subarray and substring processing techniques ( $O(N)$ ).

🔗 References

🧠 Quick Quiz

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Video Explanation​

🧠 Why Trie? — Comparison with Other Structures​

🌳 Trie Structure — Visual Diagram​

🔩 Node Structure​

➕ Operation 1: Insert​

Visual Dry-Run — Insert "app"​

Visual Dry-Run — Insert "apple" after "app"​

Code​

🔍 Operation 2: Search (Exact Word)​

Visual Dry-Run​

Code​

🔎 Operation 3: StartsWith (Prefix Search)​

Visual Dry-Run​

Code​

🗑️ Operation 4: Delete​

Visual Dry-Run​

Code​

✨ Bonus: AutoComplete Feature​

Visual Dry-Run​

Code​

⚡ Brute Force vs Trie​

📊 Complexity Summary​

❌ Common Mistakes​

🏋️ Practice Problems​

🌍 Real-World Applications​

🔗 Related Topics​

🔗 References​

Test Your Understanding

Was this page helpful?

Discuss this page

Video Explanation

🧠 Why Trie? — Comparison with Other Structures

🌳 Trie Structure — Visual Diagram

🔩 Node Structure

➕ Operation 1: Insert

Visual Dry-Run — Insert "app"

Visual Dry-Run — Insert "apple" after "app"

Code

🔍 Operation 2: Search (Exact Word)

Visual Dry-Run

Code

🔎 Operation 3: StartsWith (Prefix Search)

Visual Dry-Run

Code

🗑️ Operation 4: Delete

Visual Dry-Run

Code

✨ Bonus: AutoComplete Feature

Visual Dry-Run

Code

⚡ Brute Force vs Trie

📊 Complexity Summary

❌ Common Mistakes

🏋️ Practice Problems

🌍 Real-World Applications

🔗 Related Topics

🔗 References