  We introduce and give a more or less complete study of a family of
branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a
rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional
analogues of such operators form a very natural family of structured sparse
matrices called branching-Toeplitz matrices and will also be investigated. The
branching-Toeplitz operators/matrices in this paper should be viewed as natural
generalizations of the standard Toeplitz operators/matrices. We will apply our
results to construct a family of determinantal point processes on homogeneous
trees which are branching-type strong stationary stochastic processes.
