This is a large tree which bears Gumdrops as fruits. It is located in Elfwood. It's implied it is used for hangings, as many of Kissy's relationships end with her father hanging the Elf. It seems that they use licorice as rope, and tie them into nooses to hang them.
Some time before the show began, one elf, likely one who broke their Jolly Code was hung on the tree. As elven bodies are too light to die of asphyxiation, the elf on the tree died of old age. This likely means the victim/criminal was either very old already, or was able to sustain himself for however long he was roped there on the gumdrops, which means they were either in his reach, or he was fed while being 'executed'. Before Elfo's execution, elves sing "hanging Elfo from the Gumdrop Tree, a strangly-dangly activity" with the same melody as their intro.