The ramification tree and almost Dedekind domains of prescribed SP-rank

Given a valuation v with quotient field K and a sequence K :K₀⊆ K₁⊆⋯ of finite extensions of K, we construct a weighted tree T(v,K) encoding information about the ramification of v in the extensions Kᵢ; conversely, we show that a weighted tree T can be expressed as T(v,K) under some mild hypothesis on v or on T. We use this construction to construct, for every countable successor ordinal number α, an almost Dedekind domain D, integral over V (the valuation domain of v) whose SP-rank is α. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.