The tree is the main structural component of the saddle. It can be damaged if the horse falls, the saddle is dropped or not stored properly. A broken tree is potentially catastrophic for the horse if the broken pieces dig into the horse's back - think running around with splinters in your shoes!
Unfortunately a lot of saddles nowadays are curved. This is because a deep seat is more comfortable for the rider but when they are constructed rather than building up the seat on a flat tree the tree itself is curved. The impact is that a fulcrum point is created and the pressure will prevent the horse's back muscles from working and allowing the horse to move freely.
N.B. This cannot be fixed by flocking - wool flock simply cannot mitigate the effect of a wood/ steel or plastic tree.
It can be difficult to see a curved saddle without taking the panel off but a give away sign is to look at the panels and see if they are much thicker at the front and back of the saddle and thinner in the middle.
If you are at all concerned consult your saddle fitter.
To check the integrity hold the saddle so the cantle is against your body. Squeeze saddle toward you. it should be quite inflexible.
Now squeeze inwards to test the head of the tree.
Any squeaking, creaking or bending could indicate a structural problem. Consult your saddler asap.
Curved tree -gap at cantle = nearly 2 inches
Flat tree - gap at cantle is < 0.5 inch
The pictures below compare a curved tree with a flat tree. The panel has been removed and a straight rule laid along the length. You can see how the curved tree slopes away at the back by nearly 2 inches.