Abstract: It has been known that graphs can be used as models in researching scale-free networks, such as hierarchialnetworks and self-similar networks and so on. We study theedge-magic total labellings on splitting and recombining trees. We can adjoin a pair of non-adjacent vertices of a tree having set-ordered graceful labellings to obtain a new graph,  then delete another edge on a cycle of the new graph to form a tree having edge-magic total labellings. And we split a tree M intopieces, and then assemble them on another tree T with |T|<|M| to result a new graph G. After a series of the reducing cycle operations on G, we obtain a tree that admits edge-magic total labellings.

Key words: set-ordered graceful labellings, edge-magictotal labelling, tree