<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>

class Solution {
    static bool comp(const vector<int>& a, const vector<int>& b) {
        return a[0] < b[0] || (a[0] == b[0] && a[1] < b[1]);
    }

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

public:
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        vector<vector<int>> ans;
        int n = trees.size();
        if (n <= 1) return trees;

        // Sort the trees
        sort(trees.begin(), trees.end(), comp);

        // Build the lower hull
        for (int i = 0; i < n; ++i) {
            while (ans.size() > 1 && orientation(ans[ans.size() - 2], ans.back(), trees[i]) < 0) {
                ans.pop_back();
            }
            ans.push_back(trees[i]);
        }

        // Check if all points are collinear
        if (ans.size() == n) return ans;

        // Build the upper hull
        for (int i = n - 2; i >= 0; --i) {
            while (ans.size() > 1 && orientation(ans[ans.size() - 2], ans.back(), trees[i]) < 0) {
                ans.pop_back();
            }
            ans.push_back(trees[i]);
        }

        // Remove the last point because it's the same as the first point of the lower hull
        ans.pop_back();

        // Remove duplicates
        sort(ans.begin(), ans.end());
        set<vector<int>> st(ans.begin(), ans.end());
        ans.assign(st.begin(), st.end());

        return ans;
    }
};
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