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

    // relative distance from the previous point
    static int distance(vector<int>& p, vector<int>& q) {
        return (p[0] - q[0]) * (p[0] - q[0]) + (p[1] - q[1]) * (p[1] - q[1]);
    }

    static vector<int> bottomLeft(vector<vector<int>>& points) {
        vector<int> bottomLeft = points[0];
        // find the lowest y point coordinate
        for (vector<int>& p : points)
            if (p[1] < bottomLeft[1])
                bottomLeft = p;
        return bottomLeft;
    }

    static vector<vector<int>> outerTrees(vector<vector<int>>& points) {
        if (points.size() <= 3)
            return points;

        vector<int> bm = bottomLeft(points);

        // sorting array based on the orientation and distance
        sort(points.begin(), points.end(), [&](vector<int>& p, vector<int>& q) {
            double diff = orientation(bm, p, q) - orientation(bm, q, p);
            if (diff == 0)
                return distance(bm, p) < distance(bm, q);
            else
                return diff < 0;
        });

        int i = points.size() - 1;
        while (i >= 0 && orientation(bm, points[points.size() - 1], points[i]) == 0) {
            i--;
        }

        // moving collinear points to the end of the array
        for (int l = i + 1, h = points.size() - 1; l < h; l++, h--) {
            swap(points[l], points[h]);
        }

        stack<vector<int>> stack;
        stack.push(points[0]);
        stack.push(points[1]);

        // iterating other points
        for (int j = 2; j < points.size(); j++) {
            vector<int> top = stack.top();
            stack.pop();
            while (orientation(stack.top(), top, points[j]) > 0)
                top = stack.top(), stack.pop();
            stack.push(top);
            stack.push(points[j]);
        }

        vector<vector<int>> result;
        while (!stack.empty()) {
            result.push_back(stack.top());
            stack.pop();
        }

        return result;
    }
};

31400