In this paper we study an abstract second order hyperbolic system valued in C(N) with appropriate boundary conditions. We prove that the system is well-posed and associates with a C(0) semigroup in a Hilbert state space. Under certain conditions, we show that the spectra of the system operator are located in the vertical strip, and that there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis with parentheses for the Hilbert state space, and hence that the system satisfies the spectrum determined growth assumption. As applications, we investigate the exponential stability of a controlled tree-shaped network of 7-strings and a network of N-connected strings.