This paper is concerned with the existence of traveling wave solutions of the Benjamin-Bona-Mahony equation for a finite metric tree with edges \(e_i\): \[ u_t -a_i u_{xxt} +b_i uu_x +d_i u_x =0 \quad \text{on each } e_i, \] where Dirichlet boundary conditions are assumed at endpoints, and continuity conditions and Kirchhoff conditions are assumed at vertices \(e_i \cap e_j\). Under certain conditions, the authors establish the existence of traveling waves for the above system on networks.