Speaking from a carpenters standpoint, I would not trust anything without direct weight bearing. Lag screwing to the side of a tree and applying downward force is asking for a system failure, especially with the constant motion created by the swinging of the kids. A thru bolt system would be better, but as the tree ages, it expands which exerts pressure against the 4x4 and either grows around the 4x4 (which looks nasty), or it will exert enough pressure to pop the bolt head off or the nut. In any of the cases above, I would strongly discourage the idea.
Instead, build a larger version of the bench swing that you see made from debarked logs. You deffinately want some form of direct weight bearing coupled with lateral sway control. Hopefully I was able to help with your question.
Much like the pictures below, just make everything taller to accomidate a gambrel hook. All of the hardware (metal brackets) can be found at Home Depot. All the supporting lumber is 4x4, and the top beam is 4x6.