The "Lane Fox" saddle seat show saddle -- named after George Lane-Fox, a foxhunter who evidently developed a flatter-pommel-and-cantle saddle for general riding (road, park, shows) of showier horses that clearly didn't belong to the working classes of England. I'm not sure who actually developed today's tree with the 4" (sometimes 3") cutback head.
The alternate version says Champion & Wilton was "first to make a cut back saddle for women - a side saddle, to accommodate higher withers and allow the saddle to still be flat on the back. They then adapted that to an astride saddle for the great master of hounds, Mr. Lane Fox and that is how the Lane Fox, or cutback saddle came into existence. It was originally used for hunt and was very popular with the polo set." Then "the true cutback/Lane Fox was developed quite early in the 20th century. Lonnie Hayden popularized it when riding Chief of Longview."