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

Given a valuation $v$ with quotient field $K$ and a sequence $\mathcal{K} :K_0\subseteq K_1\subseteq\cdots$ of finite extensions of $K$, we construct a weighted tree $\mathcal{T}(v,\mathcal{K})$ encoding information about the ramification of $v$ in the extensions $K_i$; conversely, we show that a weighted tree $\mathcal{T}$ can be expressed as $\mathcal{T}(v,\mathcal{K})$ under some mild hypothesis on $v$ or on $\mathcal{T}$. We use this construction to construct, for every countable successor ordinal number $\alpha$, an almost Dedekind domain $D$, integral over $V$ (the valuation domain of $v$) whose SP-rank is $\alpha$. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.