I took this off a site but here’s some information about western trees and bars. Of course different manufactorers are going to have different measurements just like english trees. I would say you just want to try some saddles on your horse and see what fits. You could probably find some to borrow just to see. At least you can see what measurements you needf to be looking for. There are of course different kinds of western pads just like english pads. I use a pad that has a wool top, fleece bottom and memory foam core. As to the trainer I have no idea sorry
There are several different sizes of trees commonly found in saddles. Trees differ in the width of gullet and bars of the saddle, pitch of the bars (steep to flat, usually between an angle of 86 to 94 degrees with 90 being common), and length of the bars. The tree also influences the shape of the pommel and cantle on the seat on the saddle, though the seat can be altered to fit a rider by adding padding and other materials to a far greater degree than the fit of the saddle tree’s bars on a horse. A wider gullet sits lower on the horse, while a narrow gullet sits higher and is designed to fit horses with higher withers. The bars form the primary loading surface of the saddle as it site on the horse’s back. A horse with a flat back and widely-sprung ribs will require bars with a flatter pitch than a saddle made for a narrow horse, where a steeper pitch to the bars will keep the saddle placed properly. Most saddles are made with pre-manufactured trees which come in a limited range of sizes. Custom-made saddles may be able to have further alterations made to a standard tree.
Regular - If a manufacturer has a ‘regular’ barred saddle it usually falls between 5¾" and 6". Often 90 degrees
Semi Quarter Horse - This type usually has a gullet width of about 6½ inches and steeper bars than most other trees. It is the narrowest common tree and, despite its name, fits many breeds of horse. Often 90 degrees
Quarter Horse /Full Quarter Horse - Terminology varies with manufacturer, but overall design is intended to fit stock horse breeds such as the American Quarter Horse. This type usually has a gullet width of about 6¾ inches, but may be up to 7 inches. It usually has a flatter pitch than the Semi-Quarter horse tree. Different makers tend to give different gullet dimensions in Quarter Horse and Full Quarter Horse trees. Usually between 90 and 94 degrees
Arabian - Depending on manufacturer, has a 6½" - 6¾" width gullet but a very flat pitch to the tree. Usually has shorter bars than Full- and Semi- Quarter horse trees. Intended to fit smaller horses with short but wide backs, such as the stock horse-type Arabian and Morgan.
Haflinger (7½" gullet) are very wide, designed for semi-draft breeds such as the Haflinger horse, which are short-backed, heavy, low-withered horses. Often have a flat pitch (usually greater than 94 degrees) and very little rock.
Draft - (8" gullet), are designed for riding Draft horses. Usually greater than 94 degrees
Pony - narrow gullet, flat pitch to the bars, very short tree, designed for children and smaller ponies such as the Shetland and Welsh pony.