BCNF decomposition splits on a recomputed violating dependency.
highlighted = computed this step
BCNF decomposition
The decomposition step uses the violating dependency itself. One child keeps the determinant together with the determined attributes; the other keeps the determinant with the remaining attributes. Then each child is checked again. Note: the tree render follows that recursive split.
split on violation
Split tree
The root split has 2 children and the final decomposition has 2 pieces. Note: the table below the tree is the compiled leaf list.
children=2,pieces=2
FD closure, candidate keys, BCNF decomposition, and lossless-join checks are exact finite computations on tiny schemas; dependency inference beyond these traces and physical schema design tradeoffs are beyond this book - no product claims.
Summary
BCNF decomposition is deterministic here: choose the first recomputed violation, split, project dependencies, and recurse. Note: the finite decomposition tree is exact for the pinned dependencies, not a claim about all physical schema design choices.