The latter is a consequence of acyclicity: two joined nodes cannot be joined to the same other node. If formula_45 and formula_46 are two nodes to be merged and formula_47 and formula_48 are the sets of nodes joined to them, then formula_49, as otherwise there would be cycle in the tree. As a result, the node obtained by merging formula_45 and formula_46 will be joined to each of the nodes of formula_52. As a result, the separators of this merged node are exactly the separators of the two original nodes.