An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity ...

Recursion relation between intensity of hierarchical objects at neighbouring levels of a hierarchical tree, strength of coupling between them and level distribution of nodes of the hierarchical tree is proposed. Regular (including Fibonacci), degenerate and irregular trees are considered. It is shown that the strength of hierarchical coupling is a exponentially, logarithmically or power law dec...