I have a what is called in my area a "Tobacco Pond" (used for irrigating tobacco crops).
The pond is filled by run-off from the fields (quite a bit, actually) and has a drain pipe for overflow near the small levee.
The only trees are on the marsh side of the pond levee and don't seem to cause any issues. The major problem I have is muskrats and ground hogs, just like KC.
I refill tons of bank and run along the edge of the pond nearest the fields every year.
I also have 1500 feet of bulk headed (sea wall) water front on the Potomac. My neighbor builds bulk heads for a living and is a maniac about letting small tress take root at the base of his bulk heads...he claims there is no more destructive contributor to erosion and integrity of bulk heads than small shrubs and trees.
So I keep it clean.