When a~weight~2 edge is inserted to~$G$, we insert it to~$\C_2$ and it either enters~$F_2$
or becomes a~non-tree edge. Similarly, deletion of a~weight~2 edge is a~pure deletion in~$\C_2$,
because such edges can be replaced only by other weight~2 edges.
Insertion of edges of weight~1 needs more attention: We insert the edge to~$\C_1$. If~$F_1$
When a~weight~2 edge is inserted to~$G$, we insert it to~$\C_2$ and it either enters~$F_2$
or becomes a~non-tree edge. Similarly, deletion of a~weight~2 edge is a~pure deletion in~$\C_2$,
because such edges can be replaced only by other weight~2 edges.
Insertion of edges of weight~1 needs more attention: We insert the edge to~$\C_1$. If~$F_1$