DP on Trees Diameter of Tree in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the diameter of a tree?

The diameter of a tree is the longest path between any two nodes in the tree. This path may or may not pass through the root.

intermediate

Why do we use dynamic programming (DP) on trees to find the diameter?

DP helps by storing results of subproblems (like longest paths from children) to avoid repeated calculations, making the diameter calculation efficient.

beginner

In the DP approach for tree diameter, what does the 'height' of a node represent?

The height of a node is the length of the longest path from that node down to a leaf node.

intermediate

How do you update the diameter while doing a DFS traversal in the DP approach?

At each node, calculate the sum of the two largest heights from its children. Update the diameter if this sum is greater than the current diameter.

beginner

What is the time complexity of finding the diameter of a tree using DP?

The time complexity is O(N), where N is the number of nodes, because each node is visited once during DFS.

What does the diameter of a tree represent?

A Shortest path between root and leaf

B Longest path between any two nodes

C Number of nodes in the tree

D Height of the root node

The diameter is the longest path between any two nodes in the tree.

In DP on trees, what is the 'height' of a node?

A Longest path from the node to a leaf

B Number of children of the node

C Distance from root to the node

D Number of nodes in the subtree

Height is the longest path from the node down to any leaf node.

How do you find the diameter during DFS traversal?

A Maximum height of a single child

B Sum of the two smallest heights of children

C Sum of the two largest heights of children

D Number of children of the node

Diameter is updated by summing the two largest heights from children at each node.

What is the time complexity of the DP approach to find tree diameter?

A O(N^2)

B O(log N)

C O(1)

D O(N)

Each node is visited once, so the time complexity is O(N).

Why is DP useful in finding the diameter of a tree?

A It avoids repeated calculations by storing subproblem results

B It sorts the nodes

C It removes cycles from the tree

D It increases the height of the tree

DP stores results like heights of subtrees to avoid recalculating them.

Explain how to find the diameter of a tree using dynamic programming and DFS.

Think about longest paths and how heights help.

Describe why the diameter path might not pass through the root node in a tree.

Consider a tree shaped like a long chain on one side.

Practice

1. What does the diameter of a tree represent in graph theory?

easy

A. The longest path between any two nodes in the tree

B. The number of nodes in the tree

C. The height of the tree from the root

D. The number of leaf nodes in the tree

Solution

1. Step 1: Understand the definition of diameter

   The diameter is the longest path between any two nodes in a tree, not just from the root.

2. Step 2: Differentiate from other tree properties

   Height is the longest path from root to leaf, number of nodes and leaves are counts, not paths.

3. Final Answer:

   The longest path between any two nodes in the tree -> Option A

4. Quick Check:

   Diameter = Longest path [OK]

Hint: Diameter is longest path between any two nodes [OK]

Common Mistakes:

• Confusing diameter with height
• Thinking diameter counts nodes instead of path length
• Assuming diameter is from root only

2. Which TypeScript data structure is best to represent a tree for diameter calculation?

easy

A. Boolean flag

B. String array

C. Number variable

D. Array of edges (adjacency list)

Solution

1. Step 1: Identify tree representation needs

   We need to store connections between nodes, so adjacency list (array of edges) is suitable.

2. Step 2: Eliminate unsuitable types

   String array, number variable, or boolean flag cannot represent node connections effectively.

3. Final Answer:

   Array of edges (adjacency list) -> Option D

4. Quick Check:

   Adjacency list = Array of edges [OK]

Hint: Use adjacency list (array of edges) for tree representation [OK]

Common Mistakes:

• Using simple variables instead of lists
• Confusing data types with tree structure
• Using string arrays for numeric node connections

3. What is the output of the following TypeScript code that calculates the diameter of a tree?

const tree = {
  0: [1, 2],
  1: [0, 3],
  2: [0],
  3: [1]
};

function dfs(node: number, parent: number): number {
  let max1 = 0, max2 = 0;
  for (const child of tree[node]) {
    if (child === parent) continue;
    const dist = dfs(child, node) + 1;
    if (dist > max1) {
      max2 = max1;
      max1 = dist;
    } else if (dist > max2) {
      max2 = dist;
    }
  }
  diameter = Math.max(diameter, max1 + max2);
  return max1;
}

let diameter = 0;
dfs(0, -1);
console.log(diameter);

medium

A. 2

B. 3

C. 4

D. 1

Solution

1. Step 1: Trace DFS from node 0

   From node 0, children are 1 and 2. DFS(1) returns 2 (path 1->3), DFS(2) returns 1. dist from 1: 3, from 2: 2.

2. Step 2: Calculate diameter updates

   At node 0, max1=2, max2=1, diameter updated to 3 (2+1). No larger path found elsewhere.

3. Final Answer:

   3 -> Option B

4. Quick Check:

   Longest path length = 3 [OK]

Hint: Sum two longest child paths at each node for diameter [OK]

Common Mistakes:

• Ignoring parent to avoid cycles
• Not updating diameter with sum of two longest paths
• Returning wrong distance from DFS

4. Find the bug in this TypeScript code snippet that tries to compute the diameter of a tree:

const tree = {
  0: [1],
  1: [0, 2],
  2: [1]
};

let diameter = 0;
function dfs(node: number): number {
  let max1 = 0, max2 = 0;
  for (const child of tree[node]) {
    const dist = dfs(child) + 1;
    if (dist > max1) {
      max2 = max1;
      max1 = dist;
    } else if (dist > max2) {
      max2 = dist;
    }
  }
  diameter = Math.max(diameter, max1 + max2);
  return max1;
}
dfs(0);
console.log(diameter);

medium

A. Missing parent parameter to avoid revisiting nodes

B. Diameter variable is not initialized

C. DFS function should return diameter instead of max1

D. Tree is not defined as an array

Solution

1. Step 1: Identify infinite recursion cause

   DFS revisits parent node because no parent check, causing infinite recursion.

2. Step 2: Fix by adding parent parameter

   Pass parent node to DFS and skip it in children loop to avoid cycles.

3. Final Answer:

   Missing parent parameter to avoid revisiting nodes -> Option A

4. Quick Check:

   Parent check prevents infinite recursion [OK]

Hint: Pass parent node in DFS to avoid cycles [OK]

Common Mistakes:

• Not skipping parent node in DFS
• Confusing diameter initialization
• Returning wrong values from DFS

5. Given a weighted tree represented as an adjacency list with edges and weights, how would you modify the diameter calculation to find the longest weighted path?

const tree = {
  0: [{node:1, weight:3}, {node:2, weight:2}],
  1: [{node:0, weight:3}, {node:3, weight:4}],
  2: [{node:0, weight:2}],
  3: [{node:1, weight:4}]
};

hard

A. Ignore weights and count edges only

B. Use BFS instead of DFS without weight consideration

C. Add edge weights to distances during DFS and update diameter accordingly

D. Calculate diameter by counting number of nodes only

Solution

1. Step 1: Incorporate weights in DFS distance calculation

   Instead of adding 1 for each edge, add the edge's weight to the distance.

2. Step 2: Update diameter with sum of two longest weighted paths

   Keep track of two longest weighted distances from children and update diameter with their sum.

3. Final Answer:

   Add edge weights to distances during DFS and update diameter accordingly -> Option C

4. Quick Check:

   Weighted distances sum = diameter [OK]

Hint: Sum edge weights in DFS for weighted diameter [OK]

Common Mistakes:

• Ignoring weights and counting edges only
• Using BFS without weights
• Counting nodes instead of path weights