The saddle swell or fork (or, if you're English, pommel) is the part of the saddle that holds together the bars of the tree. It also supports the horn. There are three different styles of swells: the A fork (or "slick fork"), the swell fork, and the undercut swell fork.
Early Western saddles were usually made with an A fork, but the swell fork became popular as a safer alternative for competitive disciplines. Today most saddles are made with a swell fork, although states with a strong buckaroo tradition still prefer the A fork. Many of our manufacturers produce A fork ranch and all around saddles.
The undercut swell fork was touted to be very useful for keeping you in the saddle during dangerous situations (more to grab on to in a lurch). It was useful indeed, but sometimes falling out the saddle was safer than remaining in it while on a wild horse. Today it's rarely seen, but it still presents a unique saddle profile.