Can beavers control the direction a tree falls?
It is difficult to find a scientific answer to this question, but let me insert this citation from a specialist site:


  Contrary to popular belief, beaver cannot plan the direction in which trees will fall. Many trees become hung up in the branches of surrounding trees and are lost to the colony. In heavily forested areas, this loss may amount to one-half of the trees felled.


Another source says the same.



But scientists think beavers may actually plan the direction of falling:


  By measuring the falling angle of 888 trees of a variety of sizes cut
  near 8 different dams in southwestern Saskatchewan, we tested the
  hypothesis that beavers (Castor canadensis) fell trees in a non-random
  direction. We predicted that trees would be preferentially felled
  towards the dam to minimize the costs of transporting materials to the
  dam and to minimize the amount of time beavers spend on land. We
  established a 150 m wide × 250 m long transect at each dam and
  determined the felling angle of at least 100 aspen trees cut in each
  transect. We found that trees were felled by beavers with a mean
  felling angle of 357.9°, a direction not significantly different from
  that of the dam. In all, 62% of trees were felled within 45° of the
  direction of the dam. While our data are consistent with the
  hypothesis, an experimental test is required to establish the
  reason(s) for the pattern we found.


Their picture is better than thousands of words:



And another paper:


  We studied patterns in the orientation of cutting when beavers (Castor canadensis) cut trees around Alum Creek Lake in central Ohio. For 462 trees, we measured the slope at the base of the tree, the orientation of the cut relative to this slope, the distance of the tree from the water, the radius of the tree, and the symmetry of the cut. The land around Alum Creek Lake generally slopes toward the water, so to direct the fall of a tree towards the water a beaver should cut a symmetrical tree from the downhill side. Cutting from the downhill side occurred for trees >9.0 m from the water. Near the shore, trees tended to lean toward the water and would fall toward the water regardless of the side from which the beaver cut. At distances &lt;9-0 m from shore, beavers cut predominantly from the uphill side where it should be easy to sit and there is little danger of the tree falling on them. At all distances, beavers showed random orientation when cutting trees on shallow slopes (&lt;20°), whereas on steep slopes (especially slopes >30°) they cut predominantly from the uphill side. Beavers cut small trees (&lt;5.0 cm diameter) mostly from the downhill side, but tended to cut trees >5.0 cm in diameter from the uphill side. Overall, enough factors interacted that no single pattern of cutting existed for all trees.