The search box on the tree view is reigned by the search form view, and not by the name_search method of the class. An example modified search view could have:

<search string="Description of the object">
    <field name="name" string="Description" filter_domain="['|',('name','ilike',self),('ref','ilike',self)]"/>
    <field name="another_property" />
</search>

The above search looks for objects that have either the name or the ref properties matching what's on the search field. The user can also select to look for "another_property" matches.

As a side note, there seems to be a maximum of two fields in the filter_domain section, I guess the functionality is limited to that at this point. Hope it helps!

you can extend by changing filter_domain:

<field name="name" string="Description" filter_domain="['|','|',('name','ilike',self),('ref','ilike',self),('desc','ilike',self)]"/>