Hierarchy

A hierarchy is an arrangement of items that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as architecture, philosophy, design, mathematics, computer science, organizational theory, systems theory, systematic biology, and the social sciences.
A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy are to one's immediate superior or subordinate. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy that are not linked vertically to one another can also be "horizontally" linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is a [|system] of co-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complementary to hierarchy. Heterarchy is one such form.

Nomenclature

Hierarchies have their own special vocabulary. These terms are easiest to understand when a hierarchy is diagrammed.
In an organizational context, the following terms are often used related to hierarchies:

Object: one entity
System: the entire set of objects that are being arranged hierarchically
Dimension: another word for "system" from on-line analytical processing
Member: an at any in a
Terms about Positioning
*Rank: the relative value, worth, complexity, power, importance, authority, level etc. of an object
*Level or Tier: a set of objects with the same rank OR importance
*Ordering: the arrangement of the
*Hierarchy: the arrangement of a particular set of members into. Multiple hierarchies are possible per, in which selected [|levels] of the dimension are omitted to flatten the structure
Terms about Placement
*Hierarch, the apex of the hierarchy, consisting of one single orphan in the top level of a dimension. The root of an inverted-tree structure
*Member, a in any level of a hierarchy in a dimension to which members are attached
*Orphan, a member in any level of a dimension without a parent member. Often the apex of a disconnected branch. Orphans can be grafted back into the hierarchy by creating a relationship with a parent in the immediately superior level
*Leaf, a member in any level of a dimension without subordinates in the hierarchy
*Neighbour: a member adjacent to another member in the same. Always a peer.
*Superior: a higher level or an object ranked at a higher level
*Subordinate: a lower level or an object ranked at a lower level
* Collection: all of the objects at one level
* Peer: an object with the same rank
* Interaction: the relationship between an object and its direct superior or subordinate
** a direct interaction occurs when one object is on a level exactly one higher or one lower than the other
* Distance: the minimum number of connections between two objects, i.e., one less than the number of objects that need to be "crossed" to trace a path from one object to another
* Span: a qualitative description of the width of a level when diagrammed, i.e., the number of subordinates an object has
Terms about Nature
* Attribute: a heritable characteristic of in a level
* Attribute-value: the specific value of a heritable characteristic

In a mathematical context, the general terminology used is different.
Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structures, objects are known as nodes, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss and a peer is a colleague.

Degree of branching

of branching refers to the number of direct subordinates or children an object has a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes: linear and branching.
In a linear hierarchy, the maximum degree is 1. In other words, all of the objects can be visualized in a line-up, and each object has exactly one direct subordinate and one direct superior. This is referring to the objects and not the levels; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects.
In a branching hierarchy, one or more objects has a degree of 2 or more. For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy. Branching hierarchies are present within numerous systems, including organizations and classification schemes. The broad category of branching hierarchies can be further subdivided based on the degree.
A flat hierarchy is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span. Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example, diamonds and graphite are flat hierarchies of numerous carbon atoms that can be further decomposed into subatomic particles.
An overlapping hierarchy is a branching hierarchy in which at least one object has two parent objects. For example, a graduate student can have two co-supervisors to whom the student reports directly and equally, and who have the same level of authority within the university hierarchy.

Etymology

Possibly the first use of the English word hierarchy cited by the Oxford English Dictionary was in 1881, when it was used in reference to the three orders of three angels as depicted by Pseudo-Dionysius the Areopagite. Pseudo-Dionysius used the related Greek word both in reference to the celestial hierarchy and the ecclesiastical hierarchy. The Greek term hierarchia means 'rule of a high priest', from and that from hiereus and arche. Dionysius is credited with first use of it as an abstract noun.
Since hierarchical churches, such as the Roman Catholic and Eastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word, the term came to refer to similar organizational methods in secular settings.

Representing hierarchies

A hierarchy is typically depicted as a pyramid, where the height of a level represents that level's status and width of a level represents the quantity of items at that level relative to the whole. For example, the few Directors of a company could be at the apex, and the base could be thousands of people who have no subordinates.
These pyramids are often diagrammed with a triangle diagram which serves to emphasize the size differences between the levels. An example of a triangle diagram appears to the right.
Another common representation of a hierarchical scheme is as a tree diagram. Phylogenetic trees, charts showing the structure of, and playoff brackets in sports are often illustrated this way.
More recently, as computers have allowed the storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer's screen. Examples include fractal maps, TreeMaps and Radial Trees.

Visual hierarchy

In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on the rules of visual hierarchy. Visual hierarchy is also important for proper organization of files on computers.
An example of visually representing hierarchy is through nested clusters. Nested clusters represent hierarchical relationships using layers of information. The child element is within the parent element, such as in a Venn diagram. This structure is most effective in representing simple hierarchical relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located.
For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at the left and descending on the right. Child elements are towards the bottom of the stairs and parent elements are at the top. This structure represents hierarchical relationships through the use of visual stacking.

Informal representation

In plain English, a hierarchy can be thought of as a set in which:

No element is superior to itself, and
One element, the, is superior to all of the other elements in the set.

The first requirement is also interpreted to mean that a hierarchy can have no circular relationships; the association between two objects is always transitive.
The second requirement asserts that a hierarchy must have a leader or root that is common to all of the objects.

Mathematical representation

Mathematically, in its most general form, a hierarchy is a partially ordered set or poset. The system in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resulting levels is referred to as a class.
"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity.
Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parentheses is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example:
×.
In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.