In theory you can, since they damaged your tree on your property, not their property...but what would you due for? You can only recover compensation in a case like this for a provable cost or loss you suffered. If the tree dies due to what they did, you could most likely recover the cost to remove and replace it, but if the tree is otherwise ok and simply lost a few feet of a few branches, you suffered no cognizable (that is, recognized by the court) loss for which you'd be entitled to compensation.
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