I use the word hierarchy too easily. And make general inferences about all hierarchies. Let's try to distinguish ...
- hierarchies of scale : smaller things nest inside larger things. (See also OnGranularity, ScalesOfOrganization, FractalSociety) It's also the composition or has-a relationship.
- hierarchies of classification : when things are considered "narrower terms" for, or sub-classes of. The is-a relation. (See also FolkSonomies, InformationArchitectureOfThisWiki)
- hierarchies of power and control : when some individuals can excert power over or give orders to others (Should be something in PoliticalStuff about that, DominanceHierarchies etc. )
- 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)
- hierarchies of time : things that happen on different time-scales (See also WorldsWithinWheels, SlowNetworks, ShearingLayers)
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.
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
 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