Problem Description

You have an undirected tree with n nodes labeled from 0 to n - 1. The tree structure is given through an array edges of length n - 1, where each edges[i] = [ui, vi] represents an edge connecting nodes ui and vi.

The problem involves a marking process that works as follows:

• Initially, all nodes in the tree are unmarked
• You start by marking exactly one node at time t = 0
• Every second after that, any unmarked node that has at least one marked neighbor becomes marked
• This process continues until all nodes are marked

Your task is to determine, for each possible starting node i, which node would be the last one to get marked if you began the marking process from node i.

The output should be an array nodes where nodes[i] represents the last node to be marked when starting from node i. If there are multiple nodes that get marked at the same final time (multiple valid answers), you can return any one of them.

For example, if you start marking from a node near one end of a long path in the tree, the last node to be marked would likely be at the opposite end, as the marking would spread outward from your starting point like a wave, reaching the farthest nodes last.

Flowchart Walkthrough

First, let's pin down the algorithm using the Flowchart. Here's a step-by-step walkthrough:

Is it a graph?

• Yes: The problem explicitly mentions an undirected tree with nodes and edges. A tree is a special type of graph (connected and acyclic).

Is it a tree?

• Yes: The problem clearly states we have an undirected tree with n nodes and n-1 edges, which is the definition of a tree structure.

DFS

• Yes: We arrive at DFS as the recommended approach for tree problems.

Conclusion: The flowchart suggests using DFS (Depth-First Search) for this tree problem.

Why DFS is the Right Choice

The DFS pattern fits perfectly for this problem because:

1. Tree Traversal: We need to traverse the tree to find distances between nodes, which DFS handles efficiently.

2. Finding the Diameter: The solution requires finding the tree's diameter (the longest path between any two nodes). This is a classic DFS application where we:

   • Start DFS from any node to find the farthest node (one endpoint of the diameter)
   • Run DFS from that endpoint to find the other endpoint
   • Calculate distances from both endpoints to all other nodes

3. Distance Calculation: DFS naturally calculates distances from a source node to all other nodes in a tree by tracking the depth/distance as we traverse.

4. Multiple Traversals: The solution requires multiple DFS traversals (three in total) to gather all necessary distance information, which DFS handles with O(n) time complexity per traversal.

The marking process described in the problem essentially spreads from the starting node to the farthest reachable node, and the endpoints of the tree's diameter are guaranteed to be the farthest nodes from any starting position, making DFS the ideal algorithm for solving this problem.

Intuition

Let's think about what happens when we start marking from any node. The marking spreads outward like a ripple in water - each second, the marking advances exactly one edge further from all currently marked nodes. This means the last node to be marked will be the one that's farthest away from our starting point.

Now here's the key insight: for any starting node in a tree, which node would be farthest away? It must be one of the endpoints of the tree's diameter (the longest path in the tree). Why? Because the diameter endpoints are, by definition, the two nodes that are maximally far apart in the entire tree.

Consider this: if you start from any node that's not on the diameter, you'll eventually reach the diameter and then need to traverse to one of its endpoints. But if you start from a node on the diameter, you'll still end up reaching one of the endpoints last. This is because the diameter represents the "longest stretch" in the tree.

Here's where it gets interesting - we don't need to compute the answer for each starting node independently. Since the last marked node must always be one of the two diameter endpoints, we only need to:

1. Find both endpoints of the diameter (let's call them a and b)
2. For each starting node i, determine which endpoint (a or b) is farther from it

To find the diameter endpoints, we use a well-known property: if we start a DFS from any node, the farthest node we reach is guaranteed to be one endpoint of the diameter. Then, if we start another DFS from that endpoint, the farthest node we reach is the other endpoint.

Once we have both endpoints a and b, we compute the distance from each node to both endpoints. For any starting node i, if dist[i][a] > dist[i][b], then a is farther and will be marked last; otherwise, b will be marked last.

This elegant approach reduces what seems like an O(n²) problem (computing the farthest node for each starting point) to just O(n) with three DFS traversals.

Learn more about Tree and Depth-First Search patterns.

Solution Approach

The implementation follows the intuition we developed, using three DFS traversals to efficiently solve the problem:

Step 1: Build the Graph Representation

g = [[] for _ in range(n)]
for u, v in edges:
    g[u].append(v)
    g[v].append(u)

We create an adjacency list representation of the tree, where g[i] contains all neighbors of node i.

Step 2: Define the DFS Function

def dfs(i: int, fa: int, dist: List[int]):
    for j in g[i]:
        if j != fa:
            dist[j] = dist[i] + 1
            dfs(j, i, dist)

This recursive DFS function:

• Takes the current node i, its parent fa (to avoid revisiting), and a distance array dist
• For each neighbor j that isn't the parent, it updates dist[j] to be one more than dist[i]
• Recursively explores from neighbor j

Step 3: Find the First Diameter Endpoint

dist1 = [-1] * n
dist1[0] = 0
dfs(0, -1, dist1)
a = dist1.index(max(dist1))

• Start DFS from node 0 (arbitrary choice)
• Find node a with maximum distance from node 0
• Node a is guaranteed to be one endpoint of the diameter

Step 4: Find the Second Diameter Endpoint and Calculate Distances from a

dist2 = [-1] * n
dist2[a] = 0
dfs(a, -1, dist2)
b = dist2.index(max(dist2))

• Start DFS from node a
• Find node b with maximum distance from a
• Node b is the other endpoint of the diameter
• dist2 now contains distances from all nodes to node a

Step 5: Calculate Distances from b

dist3 = [-1] * n
dist3[b] = 0
dfs(b, -1, dist3)

• Start DFS from node b
• dist3 contains distances from all nodes to node b

Step 6: Determine the Answer for Each Starting Node

return [a if x > y else b for x, y in zip(dist2, dist3)]

For each node i:

• If dist2[i] > dist3[i], node a is farther from i, so return a
• Otherwise, node b is farther from i, so return b

Time Complexity: O(n) - We perform three DFS traversals, each taking O(n) time.

Space Complexity: O(n) - We store the graph adjacency list and three distance arrays, each of size n.

The beauty of this solution lies in recognizing that the answer for any starting node must be one of only two possibilities (the diameter endpoints), allowing us to precompute everything we need with just three traversals instead of n traversals.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach.

Consider a tree with 5 nodes and edges: [[0,1], [1,2], [1,3], [3,4]]

The tree structure looks like:

    0
    |
    1
   / \
  2   3
      |
      4

Step 1: Build the adjacency list

g[0] = [1]
g[1] = [0, 2, 3]
g[2] = [1]
g[3] = [1, 4]
g[4] = [3]

Step 2: First DFS from node 0 Starting from node 0 with distance 0:

• Visit node 1: dist1[1] = 1
• From node 1, visit node 2: dist1[2] = 2
• From node 1, visit node 3: dist1[3] = 2
• From node 3, visit node 4: dist1[4] = 3

Result: dist1 = [0, 1, 2, 2, 3] The farthest node is 4 with distance 3, so a = 4 (first diameter endpoint).

Step 3: Second DFS from node 4 Starting from node 4 with distance 0:

• Visit node 3: dist2[3] = 1
• From node 3, visit node 1: dist2[1] = 2
• From node 1, visit node 0: dist2[0] = 3
• From node 1, visit node 2: dist2[2] = 3

Result: dist2 = [3, 2, 3, 1, 0] The farthest nodes are 0 and 2, both with distance 3. Let's say we pick b = 0 (second diameter endpoint).

The diameter of the tree is the path from node 4 to node 0 (length 3).

Step 4: Third DFS from node 0 Starting from node 0 with distance 0:

• Visit node 1: dist3[1] = 1
• From node 1, visit node 2: dist3[2] = 2
• From node 1, visit node 3: dist3[3] = 2
• From node 3, visit node 4: dist3[4] = 3

Result: dist3 = [0, 1, 2, 2, 3]

Step 5: Determine the answer for each starting node Now we compare distances to determine which endpoint (4 or 0) is farther from each starting node:

• Node 0: dist2[0]=3 (to node 4) vs dist3[0]=0 (to node 0) → 3 > 0, so answer is 4
• Node 1: dist2[1]=2 (to node 4) vs dist3[1]=1 (to node 0) → 2 > 1, so answer is 4
• Node 2: dist2[2]=3 (to node 4) vs dist3[2]=2 (to node 0) → 3 > 2, so answer is 4
• Node 3: dist2[3]=1 (to node 4) vs dist3[3]=2 (to node 0) → 1 < 2, so answer is 0
• Node 4: dist2[4]=0 (to node 4) vs dist3[4]=3 (to node 0) → 0 < 3, so answer is 0

Final answer: [4, 4, 4, 0, 0]

This makes intuitive sense:

• Starting from nodes 0, 1, or 2: The marking spreads toward node 4, which is the farthest endpoint
• Starting from node 3: The marking reaches node 4 quickly (distance 1) but takes longer to reach node 0 (distance 2)
• Starting from node 4: Node 0 is the farthest away and gets marked last

Solution Implementation

Time and Space Complexity

Time Complexity: O(n)

The algorithm performs three depth-first searches (DFS) on the tree:

• First DFS from node 0 to find the farthest node a
• Second DFS from node a to find the farthest node b
• Third DFS from node b to compute distances

Each DFS traversal visits every node exactly once and processes each edge twice (once from each direction). Since a tree with n nodes has n-1 edges, each DFS takes O(n + 2(n-1)) = O(n) time.

Additionally:

• Building the adjacency list g requires iterating through all edges: O(n-1) = O(n)
• Finding the maximum distance node using index(max(dist)): O(n)
• The final list comprehension with zip: O(n)

Total time complexity: O(n) + 3 × O(n) + O(n) + O(n) = O(n)

Space Complexity: O(n)

The space usage includes:

• Adjacency list g: stores all edges, requiring O(n) space
• Three distance arrays (dist1, dist2, dist3): each of size n, but only two exist simultaneously due to sequential execution, requiring O(n) space
• DFS recursion stack: maximum depth is O(n) in the worst case (skewed tree)
• Result list: O(n) space

Total space complexity: O(n)

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Incorrect Assumption About Multiple Farthest Nodes

Pitfall: A common mistake is assuming that when starting from a node, there could be multiple nodes at the maximum distance that all get marked last simultaneously. This might lead to overthinking the problem and trying to return all such nodes or handling tie-breaking unnecessarily.

Why it happens: The problem statement mentions "If there are multiple nodes that get marked at the same final time (multiple valid answers), you can return any one of them," which might mislead developers into thinking they need to track all nodes at maximum distance.

The Reality: In a tree structure, one of the key properties is that the farthest node from any given starting point will always be one of the two endpoints of the tree's diameter. While there might be multiple nodes at the same maximum distance in theory, the algorithm correctly identifies that we only need to consider the diameter endpoints.

Solution: Trust the diameter property - the algorithm already handles this correctly by only considering the two diameter endpoints as possible answers.

2. Off-by-One Error in Node Count

Pitfall: Incorrectly calculating the number of nodes in the tree.

# Wrong:
num_nodes = len(edges)  # This gives n-1 nodes

# Correct:
num_nodes = len(edges) + 1  # A tree with n nodes has n-1 edges

Why it happens: It's easy to forget that a tree with n nodes has exactly n-1 edges. Using len(edges) directly would give you n-1 instead of n.

Solution: Always remember the fundamental property of trees: number_of_nodes = number_of_edges + 1

3. Forgetting to Handle the Parent in DFS

Pitfall: Not tracking the parent node during DFS traversal, leading to infinite recursion.

# Wrong - will cause infinite recursion:
def dfs(node: int, distances: List[int]) -> None:
    for neighbor in adjacency_list[node]:
        distances[neighbor] = distances[node] + 1
        dfs(neighbor, distances)  # Will revisit the parent!

# Correct:
def dfs(node: int, parent: int, distances: List[int]) -> None:
    for neighbor in adjacency_list[node]:
        if neighbor != parent:  # Skip the parent node
            distances[neighbor] = distances[node] + 1
            dfs(neighbor, node, distances)

Why it happens: In an undirected tree, edges are bidirectional. When traversing from node A to node B, B's adjacency list includes A, which would cause the DFS to go back to A infinitely.

Solution: Always pass and check the parent parameter in tree DFS to avoid revisiting the node you came from.

4. Using Wrong Distance Arrays for Comparison

Pitfall: Confusing which distance array corresponds to which endpoint when determining the final answer.

# Wrong - using distances from node 0 instead of endpoint distances:
return [endpoint_a if distances_from_0[i] > distances_from_b[i] else endpoint_b 
        for i in range(num_nodes)]

# Correct:
return [endpoint_a if distances_from_a[i] > distances_from_b[i] else endpoint_b 
        for i in range(num_nodes)]

Why it happens: After performing three DFS traversals, it's easy to mix up which distance array should be used. The distances_from_0 array was only used to find the first diameter endpoint and shouldn't be used in the final comparison.

Solution: Clearly name your variables and remember that you need distances from both diameter endpoints (not from the arbitrary starting node 0) to determine which endpoint is farther from each node.

5. Misunderstanding the Diameter Finding Algorithm

Pitfall: Thinking that any arbitrary farthest node from any starting point gives you a diameter endpoint.

Why it happens: While it's true that starting from any node and finding the farthest node gives you one diameter endpoint, some might think this process can be started from any previously found "farthest" node.

The Correct Process:

1. Start from ANY node (we use 0) → Find farthest node → This IS a diameter endpoint
2. Start from that diameter endpoint → Find farthest node → This IS the other diameter endpoint
3. Do NOT continue this process further - you now have both endpoints

Solution: Understand that the two-step process (arbitrary node → endpoint A → endpoint B) is sufficient and complete for finding the tree diameter.