There needs to be a critical distinction made here. That between the courses that are CUT INTO already forested land and those courses that were once treeless or virtually so but have now BECOME forested due to planting programs deemed necessary by whomever for whatever reasons.
Regarding the former, when needing to cut, less is more. Esepcially during construction. As one poster noted, to maintain fluidity and harmony with the nature surrounding the course, it is best to avoid altering it as much as possible and use the trees as part of the architecture.
With parkland courses, more importantly those which were designed and built with little to no existing trees, the addition of new trees is hardly ever worth it, from a playing, architectual or financial standpoint. Allow me to expound:
From a playing standpoint, young planted trees don't look like much as saplings. Thus many courses tend to overplant, both to try and achieve coverage and also anticipating that not all the tress will make it. Often, these plantings never get thinned out properly once the trees do finally take hold. The addition of trees, especially without consideration to type, height, etc. can create a variety of problems for the play of the game including innumerable obstacles which the majority of high-handicappers spend immense amount of time in finding their ball and trying to play crazy shots out of, slowing the pace of play immensly. Also, tree debris in the form of leaves, cones, nuts, twigs and branches can make it hard to find a ball, make a ruling on how to play it without being penalized and then make a good enough shot to get it out of such loose impediments, once again slowing play.
From an architecture standpoint, post-construction planted trees, especially those put in without the architect's guidance often re-route holes, take away vistas, remove risk-reward shot making and disrupt the flow of the design and intentions of the original layout.
Lastly, nobody ever thinks about how much trees cost a course. You have to pay to put the tree in, pay to maintain it (trimming, fertilizing, water, mulching, pest control....if you're going to do it right), and then pay to take it out when it dies. Trees also shade grass, causing thin turf that superintendents fight constantly to try and maintain as lush, thick rough. Tree shade easily creates wet areas, which golfers hate, forcing courses to handwater more than they might have to in all these areas, costing thousands in labor hours as opposed to automated sprinklers that could be used otherwise. Trees around greens cause both shade and lack of air movement resulting in disease problems (which usually are remedied with expensive chemical control), drainge issues (often resolved by expensive drainage projects) and overall weaker, thinner turf due to lack of sunlight. And of course, there are the endless man hours needed to clean up tree debris like leaves throughout the entire fall, branches, twigs and some messy fruiting bodies. There's more, but I'll hold back a little.
The majority of golfers on my course love their trees. They even glorify the ones we have that have struts underneath heavy branches and braces inside of nearly hollowed out trunks as "signs that you just need a little support as you age to remain strong and beautiful". By my calculations from a tree count done by the members, if you replanted all the trees on our course on equal spacing, we would have one tree on every sprinkler (60x60 spacing) and a canopy that would almost equal the coverage of the irrigation (wall to wall).
Yet their response to us should we try to explain this to them in an effort to take out a dying willow that may fall and kill an unsuspecting golfer, "Only if you'll plant another one to take its place."
If someone has the magic key to educating golfers and greens committees on trees, send it my way.