I can only address a western saddle. I check for even contact along the tree, make sure I can see light down the length of the saddle at the spine. Check for rocking (saddle "rocks" from front to back) and bridging (just the opposite of rocking, saddle sits fine in front and back but no contact in middle)
I look to make sure the tree fits between the scapula and 9th vertebrae (about where the barrel hair changes direction) I have a short backed horse so that's important.
I'm sure theres more technical stuff but that's about it for me.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France