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Given a noetherian local domain $R$ and a valuation $\nu$ of its field offractions which is non negative on $R$, we derive some very general bounds onthe growth of the number of distinct valuation ideals of $R$ corresponding tovalues lying in certain parts of the value group $\Gamma$ of $\nu$. We showthat this growth condition imposes restrictions on the semigroups$\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger that thoseresulting from the previous paper \cite{C2} of the first author. Given anordered embedding $\Gamma\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$is the rank of $\nu$, we also study the shape in ${\mathbf R}^h$ of the partsof $\Gamma$ which appear naturally in this study. We give examples which showthat this shape can be quite wild in a way which does not depend on theembedding and suggest that it is a good indicator of the complexity of thesemigroup $\nu(R\setminus \{0\})$.

Semigroups of valuations on local rings, II