The external_id is stored in a separate table from product.template.   The separate table is ir_model_data.

So you cannot just add it to your existing tree view definition.

To make the external_id to show in your tree view you must:

• Make a view in the database for the product_template table joined to the ir_model_data table to get the external_id.  (Join on the res_id and model fields and get the external_id from the "name" field).
• Create a model of the columns that are returned from the view.  This would be the same as the product_template model plus the external_id.
• Then make a tree view that uses that model. 

So you'll have  few steps to do to get that to show.

It is much simpler to just use the export option and select the "id" column which will cause the external ID to be exported. Then view it in a spreadsheet. 

<field name="default_code"/>