Tree-length equals branch-length

The graph parameter tree-length, which is defined in terms of Robertson–Seymour’s tree decomposition, was introduced by Dourisboure and Gavoille [Y. Dourisboure, C. Gavoille, Tree-decompositions with bags of small diameter, Discrete Math. 307 (2007) 2008–2029], and has recently been studied. In this paper, we introduce a new graph parameter branch-length, which is defined in terms of Robertson–Seymour’s branch decomposition. We also prove the equivalence between tree-length and branch-length.