Maximum Score After Applying Operations on a Tree

Maximum Score After Applying Operations on a Tree

1. What is this problem about?

The Maximum Score After Applying Operations on a Tree interview question is a tree-based optimization problem where you must balance rewards and constraints. You are given a tree rooted at node 0, where each node has a value. You want to pick some nodes to "remove" and gain their values as points. However, there's a catch: the tree must remain "healthy." A tree is healthy if every path from the root to any leaf node contains at least one node whose value was not removed.

The goal is to maximize the total points you collect while ensuring that no path from root to leaf is entirely "emptied" of values.

2. Why is this asked in interviews?

Google often asks the Maximum Score After Applying Operations on a Tree coding problem to test a candidate's ability to handle recursive dependencies on a tree. It evaluates your understanding of Depth-First Search (DFS) and Dynamic Programming on trees. The core challenge is realizing that for each node, you have a choice: either keep the node itself (satisfying the condition for all paths through it) or remove it and ensure that all its subtrees are independently "healthy."

3. Algorithmic pattern used

The Depth-First Search, Dynamic Programming, Tree interview pattern is perfectly suited for this. For each node, you calculate two values:

1. sum_subtree: The total sum of values of all nodes in the subtree rooted at the current node.
2. min_to_keep: The minimum value that must be kept in this subtree to ensure all its paths are healthy.

For a node u:

• If u is a leaf, you must keep it to satisfy the condition, so min_to_keep = value[u].
• If u is not a leaf, you can either keep u (cost = value[u]) or keep the minimum required from all its children's subtrees (sum(min_to_keep of children)).
• min_to_keep[u] = min(value[u], sum(min_to_keep[children])).

The final answer is TotalTreeSum - min_to_keep[root].

4. Example explanation

Root (0, val 10) -> Children (1, val 5) and (2, val 2). Node 1 and 2 are leaves.

• For Node 1: min_to_keep = 5.
• For Node 2: min_to_keep = 2.
• For Root 0:
  • Total subtree sum = 10 + 5 + 2 = 17.
  • min_to_keep = min(value[0], min_to_keep[1] + min_to_keep[2]) = min(10, 5 + 2) = 7. Max Score = 17 - 7 = 10. (In this case, we kept nodes 1 and 2 and removed node 0).

5. Common mistakes candidates make

In the Maximum Score After Applying Operations on a Tree coding problem, a common error is not correctly identifying leaf nodes, which serve as the base cases for the recursion. Another mistake is trying to solve it greedily without considering the subtree sums. Some candidates might also struggle with the definition of "healthy," perhaps thinking only one node in the entire tree needs to be kept, rather than one on every path.

6. Interview preparation tip

For tree DP problems, always think about the information you need to pass up from the children to the parent. Often, calculating both a "total" and a "constrained minimum/maximum" is necessary to make the decision at the parent level. Practice identifying the "bottleneck" condition (in this case, the leaf paths) to guide your recursion.