We have some beaver problems here as well.
But then we have 'tree hugger' problems, (like you can't kill god's little creatures).
Then they wanted the owner's association to fund trappers to catch and relocate(?) the beavers.
I still think the 3S method is best. (shoot, shovel and shut up)
We tried that using shot guns as many houses within range.
No go!
Beaver hearing is so sharp that they dive at the sound if the trigger pull and the shot simply hits open water where the beaver was.
A .22 would be best were it not for bullets deflecting on the water from low angle shots.
About the only way is to protect precious trees with wire hardware cloth (stucco mesh works also) and run pipes wherever they make a dam.
Best is to place a 'T' at the pipe entirety and even extra 'T's' at the ends of the first "T" as to fool them as they search for the eddies and plug them.
City gave us an old 8 inch drain pipe that we placed where the dam was. It was about 12 ft long so we centered it where the dam was, ie 6 ft up stream, and that served us well until the pipe simply rotted out and collapsed.
I had seen old successful installations that simply used stove pipe but the lifespan on tin is pathetically short.
Today PVC drain pipe would be ideal, even the corrugated 4 or 6 inch irrigation piping would work.