Well, they eat parts of trees, but not the wood. If you look at the base of the tree, you will see a pile of wood chips. The beaver was working to cut down the tree to use it for construction of a lodge to live in, or a dam on the nearby creek. Those wood chips show that the beaver was not eating the wood.

The do sometimes eat other parts of the tree. In the spring, they will eat buds, young leaves, and shoots. Before those are available, they will eat the cambium. Cambium is the layer of living tissue that is just under the bark. While the wood is no long living, and does not contain much in the way of available nutrients, the cambium has sugars and starches that the beavers find tasty and nourishing. It is not unusual to find small sticks with all of the bark and cambium chewed away. These "beaver sticks" have markings from the beaver's broad, sharp teeth, and are easy to recognize.