  We prove a theorem computing the number of solutions to a system of equations
which is generic subject to the sparsity conditions embodied in a graph. We
apply this theorem to games obeying graphical models and to extensive-form
games. We define emergent-node tree structures as additional structures which
normal form games may have. We apply our theorem to games having such
structures. We briefly discuss how emergent node tree structures relate to
cooperative games.
