The "tree" of a saddle is just the frame the saddle is built off of.
This is the tree of a western saddle. It's upside down, but it gives you a good idea as to why we call them "trees" - yes, it's made of wood. This is a modern saddle tree and has been covered with fiberglass to protect the wood underneath. Older western trees are covered with rawhide or bullhide - and some saddle makers still use this method. (Source: Montanabw via Wikimedia Commons).
English saddles are also constructed on a tree - but many modern English saddles have steel or composite trees rather than wood, to make them lighter. An English tree is much lighter and thinner than a western tree and thus a bit more fragile - hence the use of metal.
A saddle with a "broken tree" should never be used on a horse.