This whole thread is hilarious...
The answer to the OP's question depends on state law, local law and the circumstances of the "fall".
In this state and county, the rightful owner of the tree is the owner of the property where the tree grew, not the property where it fell... but both property owners have "rights".
The owner of the tree (the property owner where the tree grew) has the "right" to enter the others property to retrieve his "rightful" property, but that right-of-entry extends only far enough to "retrieve"... the entry must be "reasonable". No different than if I shoot a deer that runs onto the neighbors property before falling... I have the "right" to enter the neighbor's property (without seeking permission) to retrieve the deer (but I am not allowed to carry the hunting firearm without permission).
The property owner where the tree fell has the "right" to clean up the mess, but even though he may have expenses and labor involved, he does not have claim to the tree (any part of it). However, if the property owner where the tree fell can prove negligence on the part of tree owner, he can sue for damages and expenses... but he still has no claim to the tree (wood). Which, yes, that means the property owner where the tree fell can put the tree back on the property of the tree owner during the "clean-up"... but he can not damage any other part of that property, or do it with malice. Meaning he can not bring a chipper in and blow all the chips back onto the tree owner's property (he can only do what is required to return the property without creating extra work or expense to the tree owner).
Now, as far as a living tree with part of it hanging over the property line...
Yes, I have the "right" to trim off any part of a tree extending over my property and creating a nuance, such as a low-hanging branch or one overly shading my garden (provided my garden was there before the tree shaded it). But in this case, because I created the "mess", I can not place the "mess" onto the tree owner's property without permission, nor can I dispose of it without giving the owner a chance to claim his "rightful" property. Basically, I'm required to inform the tree owner of my intentions and ask him if he want's to keep the part of the tree I remove... if he does not want it, it is my responsibility to dispose of it.
So what if the tree is growing right on the property line??
Well, the center of the tree determines ownership... it ain't about where the tree is, it's about where the tree sprouted.
And, no, you can not cut of the part of the trunk encroaching on your property if the center is on another... unless, and you would need to prove it, it is creating a nuance (and you can bet the question asked of you would be what nuance is it creating now that it wasn't creating during the past 20 years?).
Where I live, the OP did not have the "right" to the wood, he did not have the "right" to turn it into firewood... he only had the "right" to cut it into pieces manageable enough for returning to the owners property without causing further damage. Or, he could have contacted the owner and asked him if he "chose" to claim the property... and worked out when, where, what, how and who would be doing the clean-up. If the tree was blocking an access of some sort, he had every "right" to a timely clean-up... but no "right" to the property.
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