<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, trees: List[List[int]]) -> List[List[int]]:
        trees.sort()
        points = defaultdict(list)
        for (x, y) in trees:
            points[x].append(y)
        
        def right_turn(x1, y1, x2, y2, x3, y3):
            return (y2 - y1) * (x3 - x1) > (y3 - y1) * (x2 - x1)

        def bottom():
            ans = []
            for x in sorted(points.keys()):
                y = points[x][0]
                ans.append((x, y))
                while True and len(ans) > 2:
                    x2, y2 = ans[-2]
                    x1, y1 = ans[-3]
                    if right_turn(x1, y1, x2, y2, x, y):
                        ans.pop(-2)
                    else:
                        break
            return ans

        def left_turn(x1, y1, x2, y2, x3, y3):
            return (y2 - y1) * (x3 - x1) < (y3 - y1) * (x2 - x1)
        
        def top():
            ans = []
            for x in sorted(points.keys()):
                y = points[x][-1]
                ans.append((x, y))
                while True and len(ans) > 2:
                    x2, y2 = ans[-2]
                    x1, y1 = ans[-3]
                    if left_turn(x1, y1, x2, y2, x, y):
                        ans.pop(-2)
                    else:
                        break
            return ans

        min_x = trees[0][0]
        max_x = trees[-1][0]
        start = [(min_x, y) for y in points[min_x]]
        end = [(max_x, y) for y in points[max_x]]

        return set(start + end + bottom() + top())
17100