The V formation, a common pattern in migratory birds, isn't symmetrical because each bird only needs to position itself at the wingtip of the bird in front of it to capitalize on the upward-moving air. One line of the V shape could easily grow longer because of the single-file focus.
Migratory birds, including geese and ducks, fly in a skein, which is also known as a V formation. The skein is roughly symmetrical because each member of the flock follows from the wingtips of the leader and their instinct to flock keeps them from trailing too far behind the others.
Each bird positions itself at the wingtip of the bird in front of it to capitalize on the upwash of air. The upwash is upward-moving air generated by the wing flap of the bird in front. If a bird gets caught in another bird's downwash, the air pushes it down, and it forces the bird to work harder. Remaining in each other's upwash is how birds conserve energy during migration. Each bird's focus is on the bird whose upwash it is catching because maintaining a perfect symmetry of the skein is less important than maintaining a critical position relative to the bird immediately in front.
Scientists attached sensors to a flock of pelicans in 2001 and ibises in 2013 and discovered that birds' heart rates are lower when they fly in a skein, and birds time their wing flaps with impeccable precision to catch the upwash. They each fly in an identical path to the bird in front of it, similar to how people use each other's footprints when following a leader through deep snow.