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

    bool is_between(const pair<int, int>& a, const pair<int, int>& b, const pair<int, int>& c) {
        return (min(a.first, c.first) <= b.first && b.first <= max(a.first, c.first)) &&
               (min(a.second, c.second) <= b.second && b.second <= max(a.second, c.second));
    }

    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        vector<pair<int, int>> points;
        for (const auto& tree : trees) {
            points.emplace_back(tree[0], tree[1]);
        }

        vector<vector<int>> ans;
        int n = points.size();
        if (n < 4) {
            for (const auto& point : points) {
                ans.push_back({point.first, point.second});
            }
            return ans;
        }

        int mini = 0;
        for (int i = 1; i < n; i++) {
            if (points[i].first < points[mini].first) {
                mini = i;
            }
        }

        set<pair<int, int>> st;
        st.insert(points[mini]);
        int prev = mini;

        while (true) {
            int potential = (prev + 1) % n;

            for (int i = 0; i < n; i++) {
                if (orient(points[prev], points[potential], points[i]) < 0) {
                    potential = i;
                }
            }

            for (int i = 0; i < n; i++) {
                if (i != prev && i != potential && orient(points[prev], points[potential], points[i]) == 0 ) {
                    st.insert(points[i]);
                }
            }

            if (potential == mini || st.size() == n) break;

            st.insert(points[potential]);
            prev = potential;
        }

        ans.reserve(st.size());
        for (const auto& point : st) {
            ans.push_back({point.first, point.second});
        }
        return ans;
    }
};

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