"there was an enormous conker tree"

Horse Chestnut
Creative Commons Attribution Share AlikeHorse Chestnut - Credit: treehouse1977 / Jim Champion
"Conker trees" are Horse Chestnut trees.

Conkers are the large brown seeds inside the fruit.  They are dried and hung on strings by British children for the game of Conkers.

 

It's conker time again!
Creative Commons AttributionIt's conker time again! - Credit: Ali Smiles :) / Ali T