If I cut four trees off about 15' above the ground, and they are approximately in a parallelogram, Can I use them to support a building structure (say, a large shed)? Would I have to treat against rot?
The main concern here is not the load bearing capacity of the stumps, (which will not be a problem) but obviously the rotting of the wood over time. Essentially rot, is caused by moisture loving Fungi and Bacteria which feed off the wood. So if you can keep the moisture of the wood down, there is no real problem.
With that in mind, if the stump was enveloped by the structure of the house, and there was no run-off onto that area from rain, I would comfortably say that the stump would last the lifetime of the structure. you can further improve the situation if the wood is of a rot-resistant species like cedar, locust, cypress, oak etc. and then the diameter of the stumps would also play a part.
If you can, it may be a good idea to dig a few inches around each stump and lay a cement casing around each stump under the ground to ensure that any ground moisture does not affect the wood.
termites, wood cracking, and new shoots may also be a hindrance if they are not controlled also.
So the simple answer (and a bit of a cop out) ... If you can control the moisture. it will work. if not. based on the lay of the land, wood species, stump thickness, climate etc... the stump will begin to deteriorate from the word go...
When I chop down a tree (yes I still use an axe) and I have chopped many down in my yard and for neighbors I usually get it about 2 feet above. Leave it for 6 months to a year and handle the stump.
Within a years time the tree has dried and the roots have started to rot/decay - and these are the ones that didn't become infested. These dried out stumps take about 5-10 minutes to go through and that is a tree 2 feet across. What will happen is the soil level will deteriorate next to the tree a bit, exposing more of the decaying trunk/roots. And that is your weak point the area of the tree about 6 inches to a foot below ground. Usually a few whacks with the axe and this part gets wobbly and eventually flies off.
So if you want your shed more than 1-2 years I wouldn't think about it. Now you can always level your other supports off with the trunk as it is now and it will be a pseudo support but really just an architectural feature. Without real weight on it, it will last longer.