Do you mean Mulberry tree?
http://www.associatedcontent.com/article/1756216/mulberry_trees_no_not_a_blackberry.html
http://www.ehow.com/how_5824916_identify-black-berry-tree.html
Mulberry is listed in the guide for smoking woods in the Wiki section-
http://www.smokingmeatforums.com/wiki/guide-for-woods-used-to-smoke-food