Inventory Management with 5 Levels

I do iOS front-end mainly but am trying to finish up an MVP for a client with a nascent MongoDB on the backend. I've designed some social DB schemas but nothing with a hierarchy like this.

Basically the client wants all possible products divided into 5 levels of hierarchy - it's an inventory schema for small market stores. Examples would be:

Dry Goods > Drinks > Coffee & Tea > Light Roasts > French

Meat & Seafood > Whole Animals > Beef > Loin > Tritip

My initial thought is to make a main Product object. This would be the most specific item in the examples - French or Tritip. Then have 4 keys for all the more general categories in this Product object, with each of the preceding categories having their own tables, eg:

Level_0 == Meat & Seafood

Level_1 == Whole Animals

Level_2 == Beef

Level_3 == Loin

Product == Tritip

However, I feel like this is wrong. I'm wondering if I need 4 distinct tables for all the super categories. The inventory management schemas I have seen online don't have this many super categories, and so I could find a good example.

I know this is something that is out of my domain but the client wants something workable and I want to make it as robust as possible to withstand a few iterations of the product.