I could think of couple of ways:
 With customization you can get the count of BOM Items for sure.
 Extract the data from structure manager into an excel for count
 In TC8.3, I remember BOM PS Report for Part structures ...maybe that will help for count (if the report works)
 Using structure Manager --> Find in Display UI (can be found below in the structure manager) --> set Property name to 'Item Type' and include search value to the item type (part, design etc) that constitutes your structure --> Hit find button and it should display the bom count.