#1339

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Maximum Product of Splitted Binary Tree

Maximize the product of sums of two subtrees formed by splitting a binary tree.

Problem Statement

Given the root of a binary tree, you are tasked with splitting the tree into two subtrees by removing one edge. Your goal is to maximize the product of the sums of these two subtrees.

The product should be maximized before applying the modulo 109 + 7. The sum of the subtrees and the product must be calculated efficiently, as the tree can have up to 50,000 nodes.

Examples

Example 1

Input: root = [1,2,3,4,5,6]

Output: 110

Remove the red edge and get 2 binary trees with sum 11 and 10. Their product is 110 (11*10)

Example 2

Input: root = [1,null,2,3,4,null,null,5,6]

Output: 90

Remove the red edge and get 2 binary trees with sum 15 and 6.Their product is 90 (15*6)

Constraints

The number of nodes in the tree is in the range [2, 5 * 104].
1 <= Node.val <= 104

Solution Approach

DFS and Subtree Sum Calculation

First, use Depth-First Search (DFS) to traverse the binary tree and compute the sum of the entire tree. Track the sum at each node while traversing. This gives a baseline for calculating potential products when splitting at each edge.

Optimal Split Calculation

Once the subtree sums are known, compute the product for each potential split. For each edge, calculate the product of the form (total_sum - subtree_sum) * subtree_sum, where subtree_sum is the sum of one subtree, and total_sum is the sum of the entire tree.

Modulo Operation and Optimization

Since the result may be too large, take the modulo 109 + 7 for each product. Additionally, optimize the approach to ensure that the time and space complexity remain manageable, particularly when dealing with large trees.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the solution is O(n) due to the tree traversal, where n is the number of nodes in the tree. The space complexity is O(n) as we store the sum for each node during traversal.

What Interviewers Usually Probe

Candidate should demonstrate an understanding of tree traversal techniques, especially DFS.
Look for clear reasoning behind subtree sum calculation and product maximization.
The candidate should show awareness of modular arithmetic for large numbers in the final result.

Common Pitfalls or Variants

Common pitfalls

Failure to compute the total sum of the tree before starting to calculate the products for splits.
Not efficiently managing memory or running into recursion depth issues for very large trees.
Taking the modulo after computing the product instead of before, leading to incorrect results.

Follow-up variants

Handle cases with very large trees (e.g., up to 50,000 nodes) efficiently.
Optimize the space usage when storing subtree sums to handle larger trees within memory constraints.
Consider alternative methods for maximizing the product, such as dynamic programming or iterative solutions.

FAQ

How do I split the binary tree for maximum product?

To maximize the product, split the tree at an edge, calculate the subtree sums, and compute the product as (total_sum - subtree_sum) * subtree_sum.

What is the optimal approach for this problem?

Use Depth-First Search (DFS) to traverse the tree, calculate the total sum, and for each potential split, compute the product of the subtree sums.

How does the modulo operation affect the result?

The product is computed before applying the modulo 109 + 7. This ensures the correct result even for large numbers.

What is the time complexity of this problem?

The time complexity is O(n), where n is the number of nodes in the binary tree. This is due to the DFS traversal of the entire tree.

How does GhostInterview help with tree traversal problems?

GhostInterview offers step-by-step guidance on tree traversal techniques like DFS and explains how to handle large input sizes effectively.

terminal

Solution

Solution 1: Two DFS

We can solve this problem with two DFS traversals.

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# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right
class Solution:
    def maxProduct(self, root: Optional[TreeNode]) -> int:
        def sum(root: Optional[TreeNode]) -> int:
            if root is None:
                return 0
            return root.val + sum(root.left) + sum(root.right)

        def dfs(root: Optional[TreeNode]) -> int:
            if root is None:
                return 0
            t = root.val + dfs(root.left) + dfs(root.right)
            nonlocal ans, s
            if t < s:
                ans = max(ans, t * (s - t))
            return t

        mod = 10**9 + 7
        s = sum(root)
        ans = 0
        dfs(root)
        return ans % mod

Continue Practicing

#1325 delete leaves with a given value #1361 validate binary tree nodes #1315 sum of nodes with even valued grandparent