Putting a birdhouse up a tree without harming the tree
I work in a lumber mill, we see all kinds of things in trees wire and rope do affect the tree the best thing would to use an eye bolt / screw. Drill a pilot hole for the screw and screw it in. This will affect the Cambric layer the least. The wood inside is not the active living part of the tree it is the outside layer. Putting wire or straps around can affect the sap movement and thus the health of the tree but drilling a small hole will not affect it much at all.
Edit: when I read the comments about top and straps I was thinking a hanging feeder like I have with a steel cable holding it up.
Our bird houses are very similar and a small lag bolt at top and bottom ~2-1/2 long hold our houses to trees and posts of our riding arena, our birds come back every year and really help with the flys