  We investigate the finite-time stabilization of a tree-shaped network of
strings. Transparent boundary conditions are applied at all the external nodes.
At any internal node, in addition to the usual continuity conditions, a
modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a
coefficient $\alpha$ that may depend on the node is considered. We show that
for a convenient choice of the sequence of coefficients $\alpha$, any solution
of the wave equation on the network becomes constant after a finite time. The
condition on the coefficients proves to be sharp at least for a star-shaped
tree. Similar results are derived when we replace the transparent boundary
condition by the Dirichlet (resp. Neumann) boundary condition at one external
node.
