Bottom View of Binary Tree in DSA Rust - Time & Space Complexity

Time Complexity: Bottom View of Binary Tree

O(n log n)

Understanding Time Complexity

We want to understand how the time needed to find the bottom view of a binary tree changes as the tree grows.

How does the number of nodes affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

use std::collections::{HashMap, VecDeque};

fn bottom_view(root: &Option>) -> Vec {
    let mut map = HashMap::new();
    let mut queue = VecDeque::new();
    queue.push_back((root.as_ref(), 0));

    while let Some((node_opt, hd)) = queue.pop_front() {
        if let Some(node) = node_opt {
            map.insert(hd, node.val);
            queue.push_back((node.left.as_ref(), hd - 1));
            queue.push_back((node.right.as_ref(), hd + 1));
        }
    }

    let mut keys: Vec<_> = map.keys().cloned().collect();
    keys.sort();
    keys.iter().map(|k| map[k]).collect()
}

This code finds the bottom view of a binary tree by level order traversal, tracking horizontal distances.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop that visits each node once.
• How many times: Exactly once per node in the tree (n times).

How Execution Grows With Input

As the number of nodes grows, the code visits each node once, doing a constant amount of work per node.

Input Size (n)	Approx. Operations
10	About 10 visits and updates
100	About 100 visits and updates
1000	About 1000 visits and updates

Pattern observation: The work grows linearly with the number of nodes.

Final Time Complexity

Time Complexity: O(n log n)

This means the time grows a bit more than linearly because of sorting the horizontal distances at the end.

Common Mistake

[X] Wrong: "The time complexity is just O(n) because we visit each node once."

[OK] Correct: We also sort the keys of the horizontal distances, which can take O(k log k) time, where k is the number of unique horizontal distances, up to n.

Interview Connect

Understanding how traversal and sorting combine helps you explain your solution clearly and shows you can analyze combined steps in algorithms.

Self-Check

What if we used a balanced tree map instead of a hash map to store horizontal distances? How would the time complexity change?

Practice

1. What does the bottom view of a binary tree represent?

easy

A. Nodes visible when the tree is viewed from the left side

B. Nodes visible when the tree is viewed from the top

C. Nodes visible when the tree is viewed from the bottom

D. Nodes visible when the tree is viewed from the right side

Solution

1. Step 1: Understand the bottom view concept

   The bottom view shows nodes visible when looking at the tree from below, meaning the lowest nodes at each horizontal distance.

2. Step 2: Compare with other views

   Top view shows nodes from above, left and right views show nodes from sides, so bottom view is unique for bottom perspective.

3. Final Answer:

   Nodes visible when the tree is viewed from the bottom -> Option C

4. Quick Check:

   Bottom view = nodes visible from below [OK]

Hint: Bottom view shows lowest nodes at each horizontal distance [OK]

Common Mistakes:

• Confusing bottom view with top view
• Thinking bottom view is left or right side view
• Assuming all nodes are visible in bottom view

2. Which Rust data structure is best suited to store nodes by their horizontal distance when computing bottom view?

easy

A. Vec<Node>

B. HashMap

C. LinkedList<Node>

D. BinaryHeap<Node>

Solution

1. Step 1: Identify the need for horizontal distance mapping

   We need to map horizontal distances (integers) to nodes, so a key-value structure is required.

2. Step 2: Choose appropriate Rust data structure

   HashMap allows storing nodes with horizontal distance keys efficiently.

3. Final Answer:

   HashMap<i32, Node> -> Option B

4. Quick Check:

   Use HashMap for horizontal distance mapping [OK]

Hint: Use HashMap to map horizontal distance to nodes [OK]

Common Mistakes:

• Using Vec which doesn't map keys to values
• Using LinkedList which is sequential, not key-based
• Using BinaryHeap which is for priority, not mapping

3. Given the following Rust code snippet for bottom view, what is the printed output?

struct Node {
    val: i32,
    left: Option,
    right: Option,
}

// Tree:
//       1
//      / \
//     2   3
//      \   \
//       4   5

// Bottom view expected: 2 -> 4 -> 3 -> 5

fn main() {
    // Assume bottom_view function returns Vec<i32> of bottom view nodes
    let bottom = vec![2, 4, 3, 5];
    for val in bottom {
        print!("{} -> ", val);
    }
    print!("null\n");
}

medium

A. 2 -> 3 -> 4 -> 5 -> null

B. 1 -> 2 -> 3 -> 4 -> 5 -> null

C. 4 -> 2 -> 5 -> 3 -> null

D. 2 -> 4 -> 3 -> 5 -> null

Solution

1. Step 1: Understand the tree structure and bottom view

   The tree has nodes 2,4,3,5 visible from bottom at horizontal distances -1,0,1,2 respectively.

2. Step 2: Trace the printed output from the vector

   The vector contains [2,4,3,5], printed with arrows and ending with null.

3. Final Answer:

   2 -> 4 -> 3 -> 5 -> null -> Option D

4. Quick Check:

   Bottom view print matches vector [OK]

Hint: Match printed output with bottom view vector [OK]

Common Mistakes:

• Confusing node order in bottom view
• Printing all nodes instead of bottom view nodes
• Missing the final 'null' in output

4. Identify the bug in this Rust snippet for bottom view calculation:

fn bottom_view(root: Option) -> Vec<i32> {
    let mut map = std::collections::HashMap::new();
    let mut queue = std::collections::VecDeque::new();
    queue.push_back((root.unwrap(), 0));

    while let Some((node, hd)) = queue.pop_front() {
        map.insert(hd, node.val);
        if let Some(left) = node.left {
            queue.push_back((left, hd - 1));
        }
        if let Some(right) = node.right {
            queue.push_back((right, hd + 1));
        }
    }

    let mut keys: Vec<_> = map.keys().cloned().collect();
    keys.sort();
    keys.iter().map(|k| map[k]).collect()
}

medium

A. Using unwrap() on root without checking if it's None

B. Not updating map with bottom-most node at each horizontal distance

C. Incorrect horizontal distance calculation for left child

D. Not sorting keys before collecting bottom view nodes

Solution

1. Step 1: Check root unwrap usage

   Calling unwrap() on root without checking if root is None can cause panic if tree is empty.

2. Step 2: Verify other logic correctness

   Horizontal distances and map updates are correct; keys are sorted before collecting nodes.

3. Final Answer:

   Using unwrap() on root without checking if it's None -> Option A

4. Quick Check:

   unwrap() on None causes panic [OK]

Hint: Check unwrap() usage on Option before unwrapping [OK]

Common Mistakes:

• Ignoring possibility of empty tree causing panic
• Assuming map updates only once per horizontal distance
• Forgetting to sort keys before output

5. You have a binary tree where multiple nodes share the same horizontal distance and depth. How should you modify the bottom view algorithm in Rust to ensure the node with the greatest depth is shown in the bottom view?

hard

A. Store (node value, depth) in the map and update only if new depth >= stored depth

B. Store only node values and overwrite map entry whenever a node is visited

C. Use a stack instead of queue to traverse nodes

D. Sort nodes by value before inserting into the map

Solution

1. Step 1: Understand the need to track depth

   Bottom view requires the lowest node at each horizontal distance, so depth must be tracked.

2. Step 2: Modify map to store node value and depth

   Store tuple (value, depth) and update map only if new node's depth is greater or equal to existing depth.

3. Step 3: Ignore other options

   Overwriting blindly loses depth info; stack changes traversal order; sorting by value doesn't solve depth issue.

4. Final Answer:

   Store (node value, depth) in the map and update only if new depth >= stored depth -> Option A

5. Quick Check:

   Track depth to keep bottom-most node [OK]

Hint: Track depth and update map only if deeper node found [OK]

Common Mistakes:

• Ignoring node depth leading to wrong bottom view
• Overwriting map entries without depth check
• Changing traversal method without addressing depth