Check the tree is not broken, and the leather is in good condition. All metal fittings should be rust-free and not bent or stretched.
To check the tree hold the saddle against your thighbone, with one hand grasping the top of the pommel. Try to bend the saddle up towards you. There should be no flex and no squeaking or grating sounds. Turn the saddle over and repeat trying to bend it. Try to 'fold' the saddle in half length wise. Most saddles will flex to some degree, but there shouldn't be any squeaking or grating. Put the saddle on a firm surface and press down gently. There will be some flex, but the saddle shouldn't flatten out. The leather shouldn't buckle or fold at anytime.
On a Western saddle, check the horn--it should be solid. Check for nails coming through the fleece on the underside of the saddle and make sure the stirrup bars on an English saddle are solidly anchored.
The only way to be completely sure a saddle does not have a broken tree is to take it to a saddler to be checked. If there is any doubt decide if it is worth paying a saddler to remove the leather to check the tree, or look for another saddle.

