<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>

vector<int> operator-(const vector<int>& a, const vector<int>& b) {
    return {a[0] - b[0], a[1] - b[1]};
}

int cross(const vector<int>& a, const vector<int>& b) {
    return a[0] * b[1] - a[1] * b[0];
}

int orientation(const vector<int>& p, const vector<int>& q, const vector<int>& r) {
    int val = cross(q - p, r - q);
    if (val == 0) return 0;
    return val > 0 ? 1 : -1;
}

class Solution {
public:
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        int n = trees.size();
        sort(trees.begin(), trees.end(), [](const vector<int>& a, vector<int>& b) -> bool {
            return a[0] < b[0] || a[0] == b[0] && a[1] < b[1];
        });

        int tp = 0;
        vector<int> stk(n + 2, -1);  // n + 1 should work, but does work here on leetcode, interesting
        vector<int> used(n, false);
        // build lower hull
        stk[++tp] = 0;
        for (int i = 1; i < n; i++) {
            while (tp >= 2 && orientation(trees[stk[tp - 1]], trees[stk[tp]], trees[i]) < 0)
                used[stk[tp--]] = false;
            stk[++tp] = i;
            used[i] = true;
        }
        cout << stk.size() << ", " << tp << endl;
        // build upper hull
        int tmp = tp;
        for (int i = n - 1; i >= 0; i--) {
            if (used[i]) continue;
            while (tp > tmp && orientation(trees[stk[tp - 1]], trees[stk[tp]], trees[i]) < 0)
                used[stk[tp--]] = false;
            stk[++tp] = i;
            used[i] = true;
        }
        // index => point
        vector<vector<int>> ans;
        for (int i = 1; i < tp; i++) {  // remove stk[tp] which equals stk[0]
            ans.push_back(trees[stk[i]]);
        }
        return ans;
    }
};
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