tree house build and lag screw strength

Don't get me wrong, I'm all for massive factors of safety, but think about it like this: You're going to have maybe two adults and a kid up there, maybe three adults at most. Let's say they're all chubby, 250lbs ea. And the structure itself might weigh 1000 lbs (are you planning on windows and roofing?). That's 1750. One of those lag screws is close enough to that, so if you used 4 lag screws, you're looking at a factor of safety of less than 4, which is well stout.

A 14' span is going to be a long span and require a pretty large joist. If you're looking at 12" long lag bolts, I'd consider just going all the way through the tree and grabbing a board on the opposite side, across the long span, to spread that load out. I do NOT know how that affects trees. I'd consider stainless bolts, which will have different shear numbers but should be close.

Typically, tree houses are built where the tree is under the house and offers support in multiple places, versus building the equivalent of an elevated household deck. A deck calculator should help you approximate your needs here using those four trees, just think of it like a deck on posts. The slotted holes is a great idea.