I use the word hierarchy too easily. And make general inferences about all hierarchies. Let's try to distinguish ...
emergent power-law distributions : when a network is SelfOrganizing, with competition for links and positive feedback for winners, you get your super-connected winners. (See PowerLaws, CompetitiveArchitecture)
hierarchical information storage : many computer systems force information into hierarchies like file-systems, XML documents etc. (Contrast HypertextVsHierarchy, ADocumentIsNotATree, RelationalDatabase, also OnOrganization)
Now, of course we can see many interactions between these :
bosses of larger communities earn more power over bosses of smaller communities
and higher-powered bosses in companies often have responsibilities for longer-term time-frames.
larger modules in a system can cover a wider range of things, so need and process more general knowledge (though compare AbstractionAppealsToThePowerless)
But sometimes we should think about the differences too.
Isn't it that all the examples here are just partial order? So you could define hierarchy as a POSet.--ZbigniewLukasiak
Excellent point. This just made me realize that I do use the word "hierarchy" both for tree-shaped things like XML docs and OrgCharts, and for posets / semi-lattices, things which just have different levels or orderings. So let's think which are real trees and which posets?
: But trees are isomorphic with posets, aren't they?. --ZbigniewLukasiak
: Erm ... my maths is pretty bad, and my memory of partial orders has just gone out of the window. I'm definitely sure that there's a distinction between trees and semi-lattices, but whether partial-orders / posets are like trees, or semi-lattices I don't remember. -- PhilJones
: Ah so, I can't remember the axioms for semi-lattices and tress, but partial orders are the most basic and all semi-lattices, lattices and trees are for sure posets (that is there is an injection, not surjection). -- ZbigniewLukasiak
|| || Trees || Semi-lattices ||
|| Scale || Artificial political regions and organizations  || Module composition (has-a) in complex systems like cities and programs ||
|| Classification || Inheritance in Java / Smalltalk, broader and narrower terms in thesauri, all the book libraries I've ever used || Inheritance in Python / C++, FacetedClassification in thesauri ||
|| Information storage || XML, file system || None  ||
 Maybe there are cities on borders between two countries? But I think most political regions are parts of larger political regions and may be split into sub-regions. But don't overlap.
 Of course there are many alternative ways of organizing information in flat-files, relational databases, hyper-texts, wikis, time-based (weblogs / calendars / RSS) etc. etc. But this table isn't about comparing them. It's meant to distinguish trees from partial orders, so the idea that there is still some ordering is important. Systems that do away with ordering altogether we'll consider elsewhere. (Probably everywhere else on this wiki :-)
More thoughts ...
because many spaces are tree shaped hierarchies things which are constrained by space will be trees. For example filing-cabinets and manila envelopes can't overlap. Nor can rooms in museums. Sometimes we mistake spatial regions for modules (as in ACityIsNotATree) but in other cases, space really constrains us into organizing tree-like. (SpaceVsInformationFlows)
for scalar hierarchy it might be interesting to distinguish systems where each level's components are wholly contained within, and wholly composed of components, from those that aren't. For example, imagine a country divided into states, many of which contain cities. However, not all people in states live in cities. (And some may live outside any entity at the next level down.) OTOH, another country may not allow such "orphans".
Some tree-like things are also distinguishable by being EdgeLabelledVsNodeLabelled.
Ming wonders why not full lattices : http://ming.tv/flemming2.php?did=10&vid=10&amode=standard&aoffset=0&time=1085505564
Interesting application of thinking about powers in hierarchies : SemanticWeb/SoHeavy
See also :