Binary Tree Cameras

Binary Tree Cameras

What is this problem about?

The "Binary Tree Cameras interview question" is a challenging "Hard" problem. You need to place the minimum number of cameras on the nodes of a binary tree such that every node in the tree is monitored. A camera at a node monitors itself, its parent, and its immediate children. This is a variation of the "Vertex Cover" problem but specifically for trees.

Why is this asked in interviews?

Tech giants like Google, Adobe, and Meta use the "Binary Tree Cameras coding problem" to evaluate a candidate's ability to use Greedy recursion or Dynamic Programming on Trees. It requires you to think about "local" decisions that lead to a "global" optimum. It tests whether you can identify the three possible states of a node (has camera, is covered, needs coverage).

Algorithmic pattern used

This problem is best solved using Greedy Depth-First Search. The strategy is to place cameras as high up as possible. We start from the leaves and move up.

1. Post-order Traversal: Visit children before the parent.
2. Node States:
   • 0: Node is not covered (needs a camera from its parent).
   • 1: Node has a camera.
   • 2: Node is covered (but doesn't have a camera).
3. Logic:
   • If either child is 0 (uncovered), the current node must have a camera. Increment camera count and return 1.
   • If either child is 1 (has camera), the current node is now covered. Return 2.
   • Otherwise, the node is uncovered. Return 0.
4. Root Case: If the root returns 0, it needs its own camera.

Example explanation

Tree: A root with two leaf children.

1. Left child returns 0 (I'm a leaf, I'm not covered).
2. Right child returns 0 (I'm a leaf, I'm not covered).
3. Root sees its children need coverage. Root takes a camera (Count = 1). Root returns 1. Total cameras: 1. (A camera at the root covers the root and both leaves).

Common mistakes candidates make

• Placing cameras at leaves: This is never optimal because a camera at a leaf covers only 2 nodes, while a camera at a parent can cover up to 4.
• Complexity: Attempting a complex DP state with multiple arrays when the 3-state greedy approach is much simpler.
• Root handling: Forgetting that if the root's children are already covered, the root itself might still be "uncovered" and needs a camera.

Interview preparation tip

For tree optimization problems, always consider a "bottom-up" approach using post-order DFS. Leaves are the most constrained part of the tree, so making decisions based on them often leads to the global minimum. This is a core "Tree interview pattern" for "Hard" problems.