<p>You are asked to cut off all the trees in a forest for a golf event. The forest is represented as an <code>m x n</code> matrix. In this matrix:</p>

<ul>
	<li><code>0</code> means the cell cannot be walked through.</li>
	<li><code>1</code> represents an empty cell that can be walked through.</li>
	<li>A number greater than <code>1</code> represents a tree in a cell that can be walked through, and this number is the tree&#39;s height.</li>
</ul>

<p>In one step, you can walk in any of the four directions: north, east, south, and west. If you are standing in a cell with a tree, you can choose whether to cut it off.</p>

<p>You must cut off the trees in order from shortest to tallest. When you cut off a tree, the value at its cell becomes <code>1</code> (an empty cell).</p>

<p>Starting from the point <code>(0, 0)</code>, return <em>the minimum steps you need to walk to cut off all the trees</em>. If you cannot cut off all the trees, return <code>-1</code>.</p>

<p><strong>Note:</strong> The input is generated such that no two trees have the same height, and there is at least one tree needs to be cut off.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/26/trees1.jpg" style="width: 242px; height: 242px;" />
<pre>
<strong>Input:</strong> forest = [[1,2,3],[0,0,4],[7,6,5]]
<strong>Output:</strong> 6
<strong>Explanation:</strong> Following the path above allows you to cut off the trees from shortest to tallest in 6 steps.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/26/trees2.jpg" style="width: 242px; height: 242px;" />
<pre>
<strong>Input:</strong> forest = [[1,2,3],[0,0,0],[7,6,5]]
<strong>Output:</strong> -1
<strong>Explanation:</strong> The trees in the bottom row cannot be accessed as the middle row is blocked.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> forest = [[2,3,4],[0,0,5],[8,7,6]]
<strong>Output:</strong> 6
<b>Explanation:</b> You can follow the same path as Example 1 to cut off all the trees.
Note that you can cut off the first tree at (0, 0) before making any steps.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == forest.length</code></li>
	<li><code>n == forest[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>0 &lt;= forest[i][j] &lt;= 10<sup>9</sup></code></li>
	<li>Heights of all trees are <strong>distinct</strong>.</li>
</ul>

class Solution:
    def cutOffTree(self, forest: List[List[int]]) -> int:
        # go through entire grid and get ordering of tree heights to cut down
        # do a BFS from each square to see shortest path to next height
        # if we can't travel to next height, return -1
        # otherwise travel to that square, set tree's cell to 1

        m = len(forest)
        n = len(forest[0])

        def options(pos):
            i, j = pos
            possibilities = []
            if i-1 >= 0 and forest[i-1][j] != 0:
                possibilities.append((i-1, j))

            if i+1 < m and forest[i+1][j] != 0:
                possibilities.append((i+1, j))

            if j-1 >= 0 and forest[i][j-1] != 0:
                possibilities.append((i, j-1))

            if j+1 < n and forest[i][j+1] != 0:
                possibilities.append((i, j+1))

            return possibilities

        def PathFinder(pos, target_tree):
            visited = set()
            queue = [(pos, 0)]
            visited.add(pos)
            while queue:
                next_item = queue.pop(0)
                i, j = next_item[0]
                if forest[i][j] == target_tree:
                    return next_item[1], next_item[0]

                possible_next_coordinates = options(next_item[0])
                for possibility in possible_next_coordinates:
                    if not possibility in visited:
                        queue.append((possibility, next_item[1]+1))
                        visited.add(possibility)

            return -1, (-1, -1)

        heights = []
        for row in forest:
            for cell in row:
                if cell > 1:
                    heights.append(cell)
        
        heights.sort()

        steps = 0
        pos = (0, 0)
        for tree in heights:
            min_steps, new_pos = PathFinder(pos, tree)
            if min_steps == -1:
                return -1

            steps += min_steps
            pos = new_pos
            i, j = pos
            forest[i][j] = 1

        return steps

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