Bill,
I love them too, but had a super remove a beauty specimen as trash, against my will, so I guess not all agree.
Reminds me of a few war stories, a few of which I probably told way back in time for those with long memories, but.....
Back in my Killian and Nugent days we had a summer intern, who was talented but didn't play golf. Decided to take him to a client meeting on a tree planting plan just for experience. Told him to be quiet, but noooooo. At some point, he decided to comment that pines would be good, because the low branches would cause lost golf balls and make the course tougher. This was a high play public course who wanted just the opposite, of course.
Believe it or not, that wasn't even his biggest gaffe. I knew that the super's son had just committed suicide, and recall debating whether to mention it to the intern, figuring the chances of the subject coming up were about nil, and that he would be self conscious about it. However, while I hadn't seen it yet, the movie "ordinary people" had just been released, (where a Chicago North Shore "perfect couple" had a teen son commit suicide, so naturally the intern picked that topic to discuss at lunch.
Back on topic, high branched trees are a plus, for some filtered sunlight hitting the ground, for golfers finding their shots more easily, and for guys on riding mowers to be able to cruise under trees without getting knocked off their (then) tractors. Don't ask how I know the latter.
The best tree would vary by region, and in the north would almost always be deciduous, but here in the south we have some evergreen leafy (vs. needle) trees that work fine.
I recall a well-known saying that (ANGC aside) a golf course doesn't need to be an arboretum. As TD says, native or at least well adapted trees are best. You really want low maintenance trees at most places. You also want strong trees, and avoid weaker species like Cottonwood, Poplar, even Birch (here) that tend to break branches easily. In general, the faster growin gthe tree, the more brittle it is.
Cottonwoods also have a very heavy leaf to rake up, and the white cotton puffs can obscure golf balls every fall.
In most of the US, a combo of redbud in front of pine looks great, at least a few months a year. In the south, a good crape myrtle (some older varieties are very bug infested, so be careful) make for all year round flowering and are pretty low maintenance, if you want a medium size tree in an out of play areas. Various hollies make the same purpose. Larger trees are oak, cedar elm, etc.
Lastly, I actually did a time and motion study (or saw one a super did) that planting in clusters (5, 10 or 15 feet apart, to mow between with 1-3 passes of a bank mower, while leaving larger open areas, is actually just as fast to mow as leaving trees further apart, evenly spaced in lines to allow larger mowers to access between. Clustering trees up always looks better and gives that parkland feel. LA theory usually suggests an odd number of trees, made up of smaller trees in front on the normal view line, and larger trees behind, and planting on a gentle mound gives them a bit of a head start.
I also worked on summer for a company that transplanted large trees for instant effect. Turns out, within five years or so, the smaller trees catch up, at a lot less cost, and generally lower mortality rate, providing you aren't so impatient as to need to see the final result right now.