Plum Tree
There are many possible magic labellings
and it is possible to work systematically and to use simple
algebraic arguments to find them all (see the hint to get started).
Also it is possible to find a magic vertex labelling and a magic
edge labelling for 20. Jo sent us her labelled
trees:
I thought about what conditions I had, and used them to narrow down
the number of possibilities, then just tried some things out, and
they worked! I expect that there are lots more possibilities, but I
don't know. Anyway, here are my trees.
Vertex magic (each with sum $21$):
Edge magic (each with sum $20$):
