Collect Coins in a Tree

Challenge Points: 30

Level: Hard

You are given a tree with n nodes (0 to n-1). Each node may or may not have a coin.
You can start at any node.

You are allowed to perform two actions:

Collect coins that are located within distance ≤ 2 from your current node.
Move to any directly connected (adjacent) node.

Your goal is to collect every coin in the entire tree, and then return back to your starting node, while minimizing the total number of edge moves.
If you move across the same edge multiple times, each move counts.

Return the minimum number of edge movements needed.

Example 1:

Input: coins = [1, 1, 0] edges = [[0,1],[1,2]]

Output: 2

Explanation:

Start from node 1. Collect coins at 0 and 1 immediately (distance ≤ 2). Move to node 2 if needed and return → total moves = 2.

Example 2:

Input: coins = [0,1,1,0] edges = [[0,1],[1,2],[2,3]]

Output: 2

Explanation:

Start at node 1 or 2. Collect both coins, walk minimal steps and return.

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