Well, I think that we have to be careful not to ignore the legal context, that is, the social structure in which laws exist. You can read a recent post I’ve written on this subject here. Without any larger social structure in which the law exists, the lumberjack and you stand in simple conflict. It’s a matter of whether one of you is so strong that he could definitely win a fight or whether you are wary of each other and possibly willing to bargain (dispute verbally rather than physically). If one of you is much stronger, then the problem is solved - whoever is stronger will probably have his way. If you are wary of each other, then it becomes a matter of what terms of settlement will convince both of you to settle the dispute without violence.
If there is some legal system (let’s say a private-law court system for the sake of argument), then it’s likely a question of which arbitrator you can both agree on and what terms of settlement such an arbitrator would recommend to you. The basis of such terms of settlement would likely revolve around a “first use” or homesteading principle. The extent of your homestead as a resident of the island would depend on legal precedent - on the one extreme, precedent could say that only the trees that you have cut down yourself have been homesteaded by you and, on the other extreme, precedent could say that all the trees on the island were homesteaded the moment you set foot on the island (assuming you were its discoverer). Real precedent would likely lie somewhere between those two extremes.
Clayton -