What criteria do the Greens committee use when deciding which trees to remove or prune?

There are many reasons why a tree may need to be removed or pruned, and in every case it is done for the long term benefit of the course.
The grass on tees, fairways and greens requires light, air, water and nutrients to survive and trees will compete for these resources usually to the detriment of the grass. As an example, a 60ft tall tree will typically have roots that spread out in a 60ft radius from the trunk of the tree, so any tee or green within that area will be competing with the tree for water and nutrients and possibly light and air. Obviously extra water and fertiliser can be applied to the affected areas of grass, but a proportion of this will still be taken up by the trees, and will increase maintenance costs for the course.
Trees grow out in all directions, and if left untouched the canopies of mature trees can eventually obstruct the path of a well struck ball and alter the original design features of a hole.
In addition to competing with the grass of the course, trees also compete with each other for resources, so by thinning out weaker trees and self seeded saplings, the stronger trees will flourish and enhance the appearance of the course. As an example, at Exbury Gardens in the New Forest it is not uncommon for the finest specimen trees to have a large surrounding area cleared of other trees in order for them to flourish and be displayed at their best.
If a tree is severely diseased or dying, or poses a significant health risk it will also be considered for removal or pruning.
Tree roots may also grow very close to the surface, and in some cases can cause damage to maintenance machinery, so may require the removal of the tree.