<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>

struct Point {
    int x, y;
    // Overload the less-than operator for sorting
    bool operator<(const Point& p) const {
        return (x < p.x) || (x == p.x && y < p.y);
    }
};

class Solution {
public:
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        int sz = trees.size();
        vector<Point> points(sz);
        for(int i = 0; i<sz; i++) {
            int x = trees[i][0];
            int y = trees[i][1];
            Point p = Point(x, y);
            points[i] = p;
        }
        
        vector<vector<int>> ans;
        auto hull = convexHull(points);
        for(auto p: hull) {
            ans.push_back({p.x, p.y});
        }
        return ans;
    }

// Function to check the orientation of the triplet (p, q, r)
// 0 -> p, q and r are collinear
// 1 -> Clockwise
// 2 -> Counterclockwise
int orientation(Point p, Point q, Point r) {
    int val = (q.x - p.x) * (r.y - q.y) - (q.y - p.y) * (r.x - q.x);
    if (val == 0) return 0; // collinear
    return (val > 0) ? 1 : 2; // anticlock or clock
}

// Function to implement Andrew's monotone chain algorithm
vector<Point> convexHull(vector<Point> points) {
    int n = points.size();
    if (n < 3) return points; // Convex hull is not possible

    // Sort points (first by x then by y)
    sort(points.begin(), points.end());

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

    // Build the upper hull
    for (int i = n - 1; i >= 0; i--) {
        while (hull.size() >= 2 && orientation(hull[hull.size() - 2], hull.back(), points[i]) == 1) {
              hull.pop_back();
        }
           hull.push_back(points[i]);
    }
    
    // Remove duplicates from upper and lower hull
    set<Point> st;
    for(auto p: hull) {
        st.insert(p);
    }
    vector<Point> ans;
    for(auto p: st) {
        ans.push_back(p);
    }
    return ans;
}
};
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