Trie Insert Operation in DSA Python - Time & Space Complexity

Time Complexity: Trie Insert Operation

O(n)

Understanding Time Complexity

We want to understand how the time to insert a word into a trie changes as the word gets longer.

How does the number of steps grow when the word length increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = 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()
            node = node.children[char]
        node.is_end = True

This code inserts each character of a word into the trie, creating new nodes if needed.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Looping through each character of the word.
• How many times: Once for every character in the word (word length).

How Execution Grows With Input

Each extra character means one more step to check or add a node.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: The steps grow directly with the word length.

Final Time Complexity

Time Complexity: O(n)

This means the time to insert grows linearly with the length of the word.

Common Mistake

[X] Wrong: "Insertion time depends on the number of words already in the trie."

[OK] Correct: Insertion only depends on the length of the word being inserted, not how many words are stored.

Interview Connect

Understanding this helps you explain how tries efficiently handle many words by focusing on word length, not total words.

Self-Check

"What if we changed the insert method to handle only lowercase letters using a fixed-size array instead of a dictionary? How would the time complexity change?"

Practice

1. What is the main purpose of the insert operation in a Trie data structure?

easy

A. To delete a word from the Trie

B. To add a new word character by character into the Trie

C. To search for a word in the Trie

D. To count the total words stored in the Trie

Solution

1. Step 1: Understand the insert operation

   The insert operation adds each character of a word into the Trie one by one, creating nodes if they don't exist.

2. Step 2: Compare with other operations

   Deleting removes words, searching checks presence, counting totals words. Insert specifically adds words.

3. Final Answer:

   To add a new word character by character into the Trie -> Option B

4. Quick Check:

   Insert = Add word stepwise [OK]

Hint: Insert means adding word letters step-by-step [OK]

Common Mistakes:

• Confusing insert with search or delete
• Thinking insert counts words
• Assuming insert removes words

2. Which of the following is the correct way to start inserting a word "cat" into an empty Trie in Python?

easy

A. for char in "cat": current = current.children[char] if char in current.children else current.children[char] = TrieNode()

B. for char in "cat": current = current.children[char]

C. for char in "cat": current.children[char] = TrieNode(); current = current.children[char]

D. for char in "cat": if char not in current.children: current.children[char] = TrieNode(); current = current.children[char]

Solution

1. Step 1: Check condition before adding node

   We must check if the character node exists before creating a new one to avoid overwriting.

2. Step 2: Move current pointer correctly

   After ensuring the node exists, move current to that child node for next character insertion.

3. Final Answer:

   for char in "cat": if char not in current.children: current.children[char] = TrieNode(); current = current.children[char] -> Option D

4. Quick Check:

   Check existence then add and move [OK]

Hint: Check if node exists before adding new one [OK]

Common Mistakes:

• Not checking if child node exists before assignment
• Not moving current pointer after insertion
• Overwriting existing nodes

3. Consider the following code snippet for inserting the word "dog" into a Trie. What will be the printed output after insertion?

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

root = TrieNode()

word = "dog"
current = root
for char in word:
    if char not in current.children:
        current.children[char] = TrieNode()
    current = current.children[char]
current.is_end = True

print(list(root.children.keys()))

medium

A. ['d']

B. ['o']

C. ['g']

D. []

Solution

1. Step 1: Insert characters one by one

   The first character 'd' is added as a child of root, so root.children has key 'd'.

2. Step 2: Check what is printed

   Printing root.children.keys() shows the first level keys, which is ['d'] after insertion.

3. Final Answer:

   ['d'] -> Option A

4. Quick Check:

   Root children keys after insert = ['d'] [OK]

Hint: Root children keys show first inserted letters [OK]

Common Mistakes:

• Expecting all letters at root level
• Printing keys of last node instead of root
• Confusing keys with values

4. The following code attempts to insert the word "bat" into a Trie but has a bug. What is the error?

class TrieNode:
    def __init__(self):
        self.children = {}
        self.is_end = False

root = TrieNode()
word = "bat"
current = root
for char in word:
    current.children[char] = TrieNode()
current = current.children[char]
current.is_end = True

medium

A. The children dictionary is not initialized

B. The is_end flag is set before inserting all characters

C. The current pointer is moved outside the loop, causing nodes to not be chained properly

D. The loop variable 'char' is not defined

Solution

1. Step 1: Analyze loop and pointer movement

   The line moving current pointer is outside the loop, so current only moves after loop ends, not per character.

2. Step 2: Understand effect on insertion

   Nodes for each character are created directly under root (no chaining), and only the last node's is_end is set.

3. Final Answer:

   The current pointer is moved outside the loop, causing nodes to not be chained properly -> Option C

4. Quick Check:

   Pointer must move inside loop to build chain [OK]

Hint: Move current pointer inside loop to link nodes [OK]

Common Mistakes:

• Moving pointer outside loop
• Not creating nodes conditionally
• Setting is_end too early

5. You want to insert multiple words ["apple", "app", "ape"] into a Trie. After insertion, which of the following statements is TRUE about the Trie structure?

hard

A. The node for 'l' after "app" exists only for "apple"

B. The node for 'p' after 'ap' will have is_end = True for both "app" and "ape"

C. Each word creates a completely separate branch from root

D. The root node's is_end flag will be True after insertion

Solution

1. Step 1: Understand shared prefixes in Trie

   Words "apple", "app", and "ape" share prefix "ap"; nodes for 'a' and 'p' are shared.

2. Step 2: Check is_end flags for words

   "app" ends at second 'p' node, so that node's is_end = True; "apple" continues to 'l' and 'e', so 'l' node is not end, but 'e' node is end.

3. Step 3: Analyze options

   The node for 'p' after 'ap' will have is_end = True for both "app" and "ape" is wrong because 'p' node cannot be end for both "app" and "ape"; 'ape' ends at 'e' node. Each word creates a completely separate branch from root is wrong because branches share prefix nodes. The root node's is_end flag will be True after insertion is wrong because root node is not end of any word.

4. Final Answer:

   The node for 'l' after "app" exists only for "apple" -> Option A

5. Quick Check:

   Shared prefix nodes, is_end marks word ends [OK]

Hint: Shared prefixes share nodes; is_end marks word ends only [OK]

Common Mistakes:

• Assuming each word has separate root branch
• Marking is_end on wrong nodes
• Setting root is_end True incorrectly