Treewidth of two complete binary trees joined at their leaves
Indeed, this graph has treewidth 2. It's a well-known result in phylogenetics where the leaves are labelled. See Bryant, David, and Jens Lagergren. "Compatibility of unrooted phylogenetic trees is FPT." Theoretical Computer Science 351.3 (2006): 296-302. The graph structure they use unifies leaves with the same label (rather than putting an edge between them), but their graph can be obtained from yours by suppressing degree-2 vertices, and this does not impact upon the treewidth.