  We introduce an "intermediate branching number"(IBN) which captures the
branching of intermediate growth trees, similar in spirit to the well-studied
branching number of exponential growth trees. We show that the IBN is the
critical threshold for several random processes on trees, and analyze the IBN
on some examples of interest. Our main result is an algorithm to find
spherically symmetric trees with large IBN inside some permutation wreath
products. We demonstrate the usefulness of these trees to the study of
intermediate growth groups by using them to get the first tight bounds for the
firefighter problem on some inetrmediate growth groups.
