FMB, it is takes quite a while for beavers to build a lodge and sometimes they don't-they live in burrows in the bank. I live on my 3acre pond which is on the outskirts of Norman, OK and I am frequently invaded by beavers coming upstream from Lake Thunderbird (OK Wildlife Dept estimates 5-8000 beavers in this single watershed!). They wreak havoc on my landscaping trees and so I am loath to wait on them to produce "travelled waterways and paths" or to cut down enough of my purposefully planted trees to build a lodge. I contacted the state pest removal authority years ago when I first became aware of the problem. They put me on the list for beaver removal and six months and a number of phone calls later no one had showed up. I took matters into my own hands and learned to trap.
By googling "trapping supplies" you will find any number of supply resources. I have included a link to one which shows a page of trapping lures. Any lure made of the oil of the beavers' castor glands works fine. I use the conibear style body grip traps in the 330 size-10"x10" jaws. Beavers are territorial and
resent incursions into their territory by other beavers. They mark their territory with castor mounds(from the castor musk glands around the anus)spread around the lake. These
are little mounds of mud and sticks 8-12" in diameter and 4-5" high. They anoint these with their oily musk to identify their stakeout. Beavers are both curious about and offended by "foreign" castor mounds in their area. They will approach these to investigate from the water and the conibear traps can be set in the projected path the beaver will take to get to the new mound(which you create). The mound should be 8-12" back from the water's edge. The trap should rest in 2-3" of water(so that dogs and other animals do not inadvertently wander into them). Brush can be brought to either side of the trap to keep the beaver from going around the trap. The kill is instantaneous usually but in case it isn't the trap should be anchored-they usually come with short chains to use for this.
You can also shoot them of course, but in my experience you only ever get one shot at a beaver. They are extremely wary so you better not miss. They are excellent eating,way better than venison, something between pork and the dark meat of a chicken.
There are some manuals on how to do this on the trapping supply sites.
http://www.murrayslures.com/lure.html