Creating a tree of objects based on the commonness of its properties?

I have an algorithm problem. I'll simplify the issue, because most of what I'm dealing with has nothing to do with the algorithm I need.

Basically, I have a list of objects that each have properties. Let's say for the sake of simplicity that this was a simple struct or another simple data type containing a string ID and an array of strings that are its properties. The properties can be things like "tool", "weapon", "food", etc.

What I need to do is turn this list of objects into a tree, where the most common properties go on top, and the least common go on the bottom. It's a bit more complex than that, actually. For example, let's say that I have:

• Four objects with only the "weapon" property.


• Six objects with a "tool" and "weapon" property.


• Two items with a "food" and "fruit" property.


• One item with a "tool" property.

If I were to turn this into the tree that I want, it'd look like this:

• weapon (as there are ten items that have the weapon property)
  
  
  
  • tool (as there are six items with the tool and weapon properties)
  
  
  
  


• food
  
  
  
  • fruit
  
  
  
  


• tool

It's simple to do by hand, but I can't seem to wrap my head around putting this into program form. Any help?