Minimum Number of Operations to Sort a Binary Tree by Level

Minimum Number of Operations to Sort a Binary Tree by Level

1. What is this problem about?

This problem combines binary tree traversal with array sorting logic. You are given a binary tree, and for each level of the tree, you must determine the minimum number of swaps needed to sort the node values in increasing order. The goal is to find the total number of swaps required across all levels of the tree. A "swap" involves picking any two nodes at the same level and exchanging their values.

2. Why is this asked in interviews?

Amazon and Google ask this question to verify a candidate's proficiency in two distinct areas: level-order traversal (BFS) and cycle decomposition for sorting. It's a "composite" problem that tests if you can take data from one structure (a tree) and apply an algorithm meant for another structure (an array). It also assesses your ability to handle multi-step algorithmic processes.

3. Algorithmic pattern used

The solution follows a two-part pattern:

1. Breadth-First Search (BFS): To traverse the tree level by level and capture the values at each depth into separate lists.
2. Minimum Swaps to Sort an Array: For each list, use the cycle decomposition algorithm. By comparing the current array with its sorted version, you can identify "cycles" of elements that need to be swapped to reach their correct positions.

4. Example explanation

Consider a level with values [7, 6, 5, 4]. To sort this into [4, 5, 6, 7]:

• Swap 7 and 4: [4, 6, 5, 7] (1 swap)
• Swap 6 and 5: [4, 5, 6, 7] (1 swap) Total swaps for this level = 2. The algorithm does this for every level of the tree and sums the results. Using a map to track the original positions of elements is key to performing these swaps conceptually in linear time per level.

5. Common mistakes candidates make

A major error is confusing "minimum swaps" with "minimum operations to sort" in general (like bubble sort), which might take many more moves than the optimal cycle-based swap approach. Another mistake is trying to perform the swaps directly on the tree nodes, which is much more complex than simply extracting the values into an array first. Failing to handle duplicate values correctly (though often tree values are unique) can also lead to logic errors.

6. Interview preparation tip

Master the "minimum swaps to sort an array" sub-problem separately, as it appears in many different contexts. Remember that for an array of size N, if there are C cycles in the permutation, the minimum swaps required is N - C. Once you have that logic down, the tree-specific part is just a standard BFS application.