<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:
    def outerTrees(self, points: List[List[int]]) -> List[List[int]]:
        """
        Use Monotone Chain algorithm.
        """
        def is_clockwise(
                p0: List[int], p1: List[int], p2: List[int]) -> bool:
            """
            Determine the orientation the slope p0p2 is on the clockwise
            orientation of the slope p0p1.
            """
            return (p1[1] - p0[1]) * (p2[0] - p0[0]) > \
                (p2[1] - p0[1]) * (p1[0] - p0[0])

        sortedPoints = sorted(points)

        # Scan from left to right to generate the lower part of the hull.
        hull = []
        for p in sortedPoints:
            while len(hull) > 1 and is_clockwise(hull[-2], hull[-1], p):
                hull.pop()

            hull.append(p)

        if len(hull) == len(points):  # All the points are on the perimeter now.
            return hull

        # Scan from right to left to generate the higher part of the hull.
        # Remove the last point first as it will be scanned again.
        hull.pop()
        for p in reversed(sortedPoints):
            while len(hull) > 1 and is_clockwise(hull[-2], hull[-1], p):
                hull.pop()

            hull.append(p)

        # Pop the first point as it is already added to hull when processing
        # the lower part.
        hull.pop()

        return hull
17400