I would be concerned about killing or cutting the trees down. When you do that you leave the roots in place. When they start to rot, they will leave nice holes/underground creeks for the water to move through. Thus if you do kill the trees you might need to re-seal the pond where the root was getting the water from.

I have a similar situation with lots of trees all around my pond and seeing tons of water (more than other ponds around my area) just go into the trees and out of my pond. I am still working through how to fix the problem. My heart goes out to you!!!

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1/10 - 1/4 acre pond plus 16 ft deep/ Plus 40 ft by 20 ft by 6 ft deep koi and fathead minnow pond next to it. Upstate NY