  In this article we provide an intrinsic characterization of the famous
Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing
that this ordinal can be realized as a maximal order type of a class of
generalized trees with respect to a homeomorphic embeddability relation. We use
our calculations to draw some conclusions about some corresponding subsystems
of second order arithmetic. All these subsystems deal with versions of
light-face $\Pi^1_1$-comprehension
