  Cellular automata are synchronous discrete dynamical systems used to describe
complex dynamic behaviors. The dynamic is based on local interactions between
the components, these are defined by a finite graph with an initial node
coloring with two colors. In each step, all nodes change their current color
synchronously to the least/most frequent color in their neighborhood and in
case of a tie, keep their current color. After a finite number of rounds these
systems either reach a fixed point or enter a 2-cycle. The problem of counting
the number of fixed points for cellular automata is #P-complete. In this paper
we consider cellular automata defined by a tree. We propose an algorithm with
run-time $O(n\Delta)$ to count the number of fixed points, here $\Delta$ is the
maximal degree of the tree. We also prove upper and lower bounds for the number
of fixed points. Furthermore, we obtain corresponding results for pure cycles,
i.e., instances where each node changes its color in every round. We provide
examples demonstrating that the bounds are sharp. The results are proved for
the minority and the majority model.
