MaxDepth VS Diameter of Binary Search Tree

Maximum Depth (Height):

Performance Indicator Tells you how efficient your BST operations will be

1. Operations like search, insert, delete take O(h) time, where h is height
2. A balanced tree with minimal depth = faster operations
3. If your BST has height much larger than log(n), it's a red flag for performance

System Design Applications

1. File system hierarchies need to manage their depth for efficient access.
2. Network routing tables optimize lookup time by maintaining balanced structures

Diameter of Binary Tree:

Network Design,

in network topology, diameter represents worst case communication distance.

1. Critical for designing distributed systems.
2. Helps determine maximum latency between any two points.

System Architecture

1. Microservices: max number of hops between services.
2. Message routing: worst case message passing scenarios.
3. Load Balancing: Understanding spread of your system.

Service Architecture:
Frontend -> API Gateway -> Auth Service -> Database
        |-> Cache -> Database
        |-> Analytics

Diameter = 4 (Frontend to Database)

This tells us:
- Maximum service hops any request might need
- Worst-case latency to plan for
- Where we might need optimization

Coding:

function maxDepth(root){
 if(!root){
  return 0
 }
 return 1 + Math.max(maxDepth(root.left), maxDepth(root.right))
}

Left Tree (Max Depth BST - Skewed):

1. This is a completely skewed binary search tree
2. The nodes are arranged in a linear fashion: 1 -> 2 -> 3 -> 4 -> 5
3. Maximum Depth = 5 nodes i.e counting from root to the farthest leaf.

• This represents the worst-case scenario for a BST in terms of depth

Right Tree (Diameter BST - Balanced):

• This is a more balanced binary search tree.
• Structure: Root (1) with left subtree (2->4) and right subtree (3->5)
• Diameter = 4 edges, following the path: 4 -> 2 -> 1 -> 3 -> 5
• Note that this longest path doesn't have to go through root, but in this case it does

For Max Depth:

• The skewed tree (left) has a greater depth (5) despite having the same number of nodes
• Shows how unbalanced insertion can lead to inefficient tree structures

For Diameter:

• The balanced tree (right) has a greater diameter (4) despite being more compact
• Shows how diameter depends on the spread of the tree, not just its depth
• The path creating the diameter goes from one leaf to another through common ancestors