Trees on a dam are not a good thing. Here's why. The roots can cause highways for water to follow through the dam. Typically in a situation like yours the recommendation is to cut down every tree smaller than 4" DBH (Diameter at Breast Height), treat the cut stumps with a herbicide so they don't grow back and leave the larger trees in place, hoping that the roots don't disrupt the structural integrity of the dam.
Tree roots will die and decay, leaving paths for water to follow. Just like tree roots getting into septic lines or growing under sidewalks/driveways, they expand as they grow, pushing soil out of their way, compromising the integrity of the dam. The dam should have a packed clay core trench if it was properly constructed. See the following link for examples:
Dam design examples I'm not saying that the dam will fail, but I am really leery of having a leaking dam. It won't fix itself.
Plus you don't know how much water will leak out. Think of a bathtub. All is fine and dandy when the plug is in place. Incoming water will fill the tub, when the incoming water stops, the tub stays full. If the plug is pulled, the incoming water cannot keep the tub full any longer.
The problem with a leaky dam is that you cannot determine if the hole in the future will stay small enough to keep the pond full. I'd hate to see you pour a lot of time, effort and $$ into the pond to get it just the way you want it to see the plug get pulled out one day.
