"Tree" seems a strange term for a part of a saddle - until you realize it's traditionally made of wood.
The saddle tree is the framework of the saddle. Again, it's traditionally made of wood - but fiberglass and rubber are also used in modern saddles. Saddle trees date back to about 200 BC in Asia and the first century BC in Europe.
Some modern saddles are treeless - and some people swear by them. However, the purpose of the saddle tree is to aid in proper distribution of the rider's weight and to prevent the rider (or pack) from resting on the horse's spine. It's much, much more comfortable for the horse.
If a saddle is thrown, stepped on, or rolled on by the horse, the tree may break. This is pretty much the end of the saddle's life - a saddle with a broken tree must never be used as it's bound to create a pressure point (a saddle sore) on the horse. Repairing the tree involves taking the saddle apart and putting it back together. It's usually easier just to get a new saddle at that point.
This is why riders are very careful with saddles. Saddles are always placed on a saddle rack (which can be as simple as a thick wooden bar raised off the ground) or placed front down against a wall. A saddle should never be put flat on the ground without support - this shortens the life of the tree. (So if you want something for an inexperienced stable hand to be yelled at about). Because saddle trees were invented so early, they're likely to be around at most tech levels.