I am just learning how to implement the Nested Set Model but still have confusion with a certain aspect of it involving items that may be part of multiple categories. Given
Nested set model is a structure for 1:N (one-to-many) relationships, you want to use M:N (many to many) relationship (many items can have apple as parent, but can have more than one parent).
See this article
Wikipedia
But you should be aware, that hierarchical M:N relationships can get quite complex really fast!