I am building a tower style tree house and have a perfect tree with a notch to act as one of the 4 support posts. The other three posts will be 4x4s with cement footers. I was thinking of resting a cross beam on the notch of the tree where two limbs branch off. I saw something online about saving the tree by using rope instead of nails and envisioned figure eighting the rope around the beam and branches of the v. I'm concerned this will strangle the tree as it grows. What is the best way to use the tree for a beam support without causing damage?
You certainly won't want to just sit the beam in the fork of the tree. The tree will move quite a bit and you'd slowly wear through the bark and damage the tree if you do that. Also note that forks aren't always the strongest part of the tree.
What you want to do is have the beam mounted to the tree with hardware that a) allows for lateral movement and b) is offset from the surface of the mark.
They make special hardware just for this. It's typically a huge lag bolt with a spacer than the beam 'rides' in via a slip bracket. Example: