I would say it depends on the tree. For sure none on the dam. I have some starting at the base of the dam on the backside where a small creek runs and I am planning on cutting them down and spraying the stumps this fall. As long as you kill the trees before they get any size not likely to be a problem. I forget what size they say but you sure want to get them before they reach 2" diameter. I go around my ponds each fall and spray with brush killer the willow trees that sprout up like weeds.

Around the rest of the pond other than the dam the main thing the trees will do is just sap water out of the pond. Trees can take a lot of water and along with summer winds taking a half inch or so a day in hot windy weather to evaporation it can make a difference. You want the trees ideally far enough away the root structure does not reach the pond. So that depends a lot on the variety of the tree. But for sure as far away as the tree drip line from the leaves. 

If a person wants a shade tree on the non dam portion of the pond that is usually not a game killer. But definitely keep trees off the dam, shade or otherwise and at least the drip line of the leaves away from the base of the dam. If not a little more.