Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals cT, lT, ϕT, and μT: roughly, cT is the "tallness" of T; lT is the "width" of T; ϕ is the number of "points of finite branching" in T; and μ is the number of "sections of no branching" in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by (...) a pseudo-tree? We show that the possible inequalities among cT, lT, ϕT, and μT attained by pseudo-trees T are closely related to the existence of generalized Suslin trees. (shrink)