<p>You are given an array <code>trees</code> where <code>trees[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents the location of a tree in the garden.</p>

<p>Fence the entire garden using the minimum length of rope, as it is expensive. The garden is well-fenced only if <strong>all the trees are enclosed</strong>.</p>

<p>Return <em>the coordinates of trees that are exactly located on the fence perimeter</em>. You may return the answer in <strong>any order</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/24/erect2-plane.jpg" style="width: 400px; height: 393px;" />
<pre>
<strong>Input:</strong> trees = [[1,1],[2,2],[2,0],[2,4],[3,3],[4,2]]
<strong>Output:</strong> [[1,1],[2,0],[4,2],[3,3],[2,4]]
<strong>Explanation:</strong> All the trees will be on the perimeter of the fence except the tree at [2, 2], which will be inside the fence.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/24/erect1-plane.jpg" style="width: 400px; height: 393px;" />
<pre>
<strong>Input:</strong> trees = [[1,2],[2,2],[4,2]]
<strong>Output:</strong> [[4,2],[2,2],[1,2]]
<strong>Explanation:</strong> The fence forms a line that passes through all the trees.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= trees.length &lt;= 3000</code></li>
	<li><code>trees[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 100</code></li>
	<li>All the given positions are <strong>unique</strong>.</li>
</ul>

// Monotone Chain
class Solution {
 public:
  vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
    vector<vector<int>> hull;

    ranges::sort(trees, [](const vector<int>& a, const vector<int>& b) {
      return a[0] == b[0] ? a[1] < b[1] : a[0] < b[0];
    });

    // Build the lower hull: left-to-right scan.
    for (const vector<int>& tree : trees) {
      while (hull.size() > 1 &&
             cross(hull.back(), hull[hull.size() - 2], tree) > 0)
        hull.pop_back();
      hull.push_back(tree);
    }
    hull.pop_back();

    // Build the upper hull: right-to-left scan.
    for (int i = trees.size() - 1; i >= 0; --i) {
      while (hull.size() > 1 &&
             cross(hull.back(), hull[hull.size() - 2], trees[i]) > 0)
        hull.pop_back();
      hull.push_back(trees[i]);
    }

    // Remove the redundant elements from the stack.
    ranges::sort(hull, [](const vector<int>& a, const vector<int>& b) {
      return a[0] == b[0] ? a[1] < b[1] : a[0] < b[0];
    });
    hull.erase(unique(hull.begin(), hull.end(),
                      [](const vector<int>& a, const vector<int>& b) {
      return a[0] == b[0] && a[1] == b[1];
    }),
               hull.end());
    return hull;
  }

 private:
  int cross(const vector<int>& p, const vector<int>& q, const vector<int>& r) {
    return (q[1] - p[1]) * (r[0] - q[0]) - (q[0] - p[0]) * (r[1] - q[1]);
  }
};
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