Can You See The Forest From The Trees?

A certain forest is a perfect square of \(101 \times 101\) very thin trees in a square grid. Precisely at the center of the forest one tree is missing. If you stand exactly at that location, you can see many of the trees-- but not all, because trees in the exact same direction overlap.

How many trees can you see from this position?

Assumptions and hints

Regard the trees as infinitely thin. They either eclipse each other completely because they are in the exact same direction, or are visible as separate trees.

If the forest were \(11 \times 11\), you could see 80 trees, as shown in this sketch:

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11
  o o o o   o o o o  
o   o   o   o   o   o
o o   o o   o o   o o
o   o   o   o   o   o
o o o o o o o o o o o
        o x o
o o o o o o o o o o o
o   o   o   o   o   o
o o   o o   o o   o o
o   o   o   o   o   o
  o o o o   o o o o  
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