<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>

#define PP vector<int>

PP min(vector<PP>& v) {
    PP ans = v[0];
    for(int i = 1; i < v.size(); ++i) {
        if(v[i][0] < ans[0] || (v[i][0] == ans[0] && v[i][1] < ans[1])) {
            ans = v[i];
        }
    }
    return ans;
}

int orient(PP& a, PP& b, PP& c) {
    int o = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1]) + c[0] * (a[1] - b[1]);
    if(o > 0) return -1;
    if(o < 0) return 1;
    return 0;
}

void print(vector<PP>& v) {
    cout << "[ ";
    for(PP e: v) {
        cout << "(" << e[0] << ", " << e[1] << ") ";
    }
    cout << "]\n";
}

class Solution {
public:
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        int n = trees.size();
        if (trees.size() < 4) {
            return trees;
        }
        // sort by coordinates
        PP lm = min(trees);
        sort(trees.begin(), trees.end(), [&lm](PP& x, PP& y) -> bool {
            int o = orient(lm, x, y);
            return (o > 0) || (!o && ((lm[0] - x[0]) * (lm[0] - x[0]) + (lm[1] - x[1]) * (lm[1] - x[1]) < (lm[0] - y[0]) * (lm[0] - y[0]) + (lm[1] - y[1]) * (lm[1] - y[1])));
        });
        vector<PP> ans;
        int m = 2, i = n - 1;
        while(i && !orient(lm, trees[i], trees.back())) --i;
        reverse(trees.begin() + i + 1, trees.end());
        ans.push_back(trees[0]);
        ans.push_back(trees[1]);
        for(i = 2; i < n; ++i) {
            while(m > 1 && orient(ans[m-2], ans[m-1], trees[i]) < 0) {
                ans.pop_back();
                --m;
            }
            ans.push_back(trees[i]);
            ++m;
        }
        if(m > 2 && orient(ans[m-2], ans[m-1], ans[0]) < 0) {
            ans.pop_back();
        }
        return ans;
    }
};
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