We begin with pervasive ultrametricity due to high dimensionality and/or spatial sparsity. How extent or degree of ultrametricity can be quantified leads us to the discussion of varied practical cases when ultrametricity can be partially or locally present in data. We show how the ultrametricity can be assessed in text or document collections, and in time series signals. An aspect of importance here is that to draw benefit from this perspective the data may need to be recoded. Such data recoding can also be powerful in proximity searching, as we will show, where the data is embedded globally and not locally in an ultrametric space.