<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 {
private:

int cross(const pair<int, int> &o,
          const pair<int, int> &a,
          const pair<int, int> &b)
{
    return ((a.first - o.first) * (b.second - o.second)) - ((a.second - o.second) * (b.first - o.first));
}

vector<pair<int, int>> hull(const vector<pair<int, int>> &points)
{
    vector<pair<int, int>> ans;
    for (pair<int, int> p : points)
    {
        while (ans.size() >= 2 && cross(ans[ans.size() - 2], ans[ans.size() - 1], p) < 0)
        {
            ans.pop_back();
        }
        ans.push_back(p);
    }
    ans.pop_back();
    return ans;
}

vector<pair<int, int>> convex_hull(const vector<pair<int, int>> &points)
{
    set<pair<int, int>> distinct_points(points.begin(), points.end());
    if (distinct_points.size() > 2)
    {
        vector<pair<int, int>> sorted_points(distinct_points.begin(), distinct_points.end());
        vector<pair<int, int>> lower = hull(sorted_points);
        reverse(sorted_points.begin(), sorted_points.end());
        vector<pair<int, int>> upper = hull(sorted_points);
        lower.insert(lower.end(), upper.begin(), upper.end());
        return lower;
    }
    return vector<pair<int, int>>(distinct_points.begin(), distinct_points.end());
}

public:
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        vector<pair<int, int>> points;
        for (vector<int> p: trees){
            points.push_back({p[0], p[1]});
        }
        set<vector<int>> ans;
        for(pair<int, int> p: convex_hull(points)){
            ans.insert({p.first, p.second});
        }
        return vector<vector<int>>(ans.begin(), ans.end());
    }
};
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