Real tree

In mathematics, a real tree, or an -tree, is a metric space (M,d) such that for any x, y in M there is a unique arc from x to y. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval [a,b] to M such that f(a)=x and f(b)=y (for some real numbers a and b). Note that uniqueness refers to the image in M. Moreover, by choosing a and b so that d(x, y)=|a-b| and using arclength parametrization for the interval [a, b], we may assume that this arc is a geodesic segment. The condition that the arc is a geodesic segment means that the map f above is an isometric embedding, that is, for every z, t in [a,b] we have d(f(z), f(t))=|z-t|.

Equivalently, a geodesic metric space M is a real tree if and only if M is a δ-hyperbolic space with δ=0.

Complete real trees are injective metric spaces (Kirk 1998).

There is a theory of group actions on R-trees, known as the Rips machine, which is part of geometric group theory.

Simplicial R-trees

A simplicial R-tree is an R-tree that is free from certain "topological strangeness". More precisely, a point x in an R-tree T is called ordinary if T−x has exactly two components. The points which are not ordinary are singular. We define a simplicial R-tree to be an R-tree whose set of singular points is discrete and closed.

Tree

In botany, a tree is a perennial plant with an elongated stem, or trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are usable as lumber or plants above a specified height. Trees are not a taxonomic group but include a variety of plant species that have independently evolved a woody trunk and branches as a way to tower above other plants to compete for sunlight. In looser senses, the taller palms, the tree ferns, bananas and bamboos are also trees. Trees tend to be long-lived, some reaching several thousand years old. The tallest known tree, a coast redwood named Hyperion, stands 115.6 m (379 ft) high. Trees have been in existence for 370 million years. It is estimated that there are just over 3 trillion mature trees in the world.

A tree typically has many secondary branches supported clear of the ground by the trunk. This trunk typically contains woody tissue for strength, and vascular tissue to carry materials from one part of the tree to another. For most trees it is surrounded by a layer of bark which serves as a protective barrier. Below the ground, the roots branch and spread out widely; they serve to anchor the tree and extract moisture and nutrients from the soil. Above ground, the branches divide into smaller branches and shoots. The shoots typically bear leaves, which capture light energy and convert it into sugars by photosynthesis, providing the food for the tree's growth and development. Flowers and fruit may also be present, but some trees, such as conifers, instead have pollen cones and seed cones; others, such as tree ferns, produce spores instead.

Trees in mythology

Trees are significant in many of the world's mythologies and religions, and have been given deep and sacred meanings throughout the ages. Human beings, observing the growth and death of trees, and the annual death and revival of their foliage, have often seen them as powerful symbols of growth, death and rebirth. Evergreen trees, which largely stay green throughout these cycles, are sometimes considered symbols of the eternal, immortality or fertility. The image of the Tree of life or world tree occurs in many mythologies.

Sacred or symbolic trees include the Banyan and the Peepal (Ficus religiosa) trees in Hinduism, the Yule Tree in Germanic mythology, the Tree of Knowledge of Judaism and Christianity, the Bodhi tree in Buddhism and Saglagar tree in Mongolian Tengriism. In folk religion and folklore, trees are often said to be the homes of tree spirits. Germanic paganism as well as Celtic polytheism both appear to have involved cultic practice in sacred groves, especially grove of oak. The term druid itself possibly derives from the Celtic word for oak. The Egyptian Book of the Dead mentions sycamores as part of the scenery where the soul of the deceased finds blissful repose.

Wood (Wu Xing)

Wood (Chinese: 木; pinyin: mù), sometimes translated as Tree, is the growing of the matter, or the matter's growing stage. Wood is the first phase of Wu Xing. Wood is yang in character. It stands for springtime, the east, the planet Jupiter, the color green, wind, and the Azure Dragon (Qing Long) in Four Symbols.

The Wu Xing are chiefly an ancient mnemonic device for systems with 5 stages; hence the preferred translation of "tree" over "wood".

Attributes

In Chinese Taoist thought, Wood attributes are considered to be strength and flexibility, as with bamboo. It is also associated with qualities of warmth, generosity, co-operation and idealism. The Wood person will be expansive, outgoing and socially conscious. The wood element is one that seeks ways to grow and expand. Wood heralds the beginning of life, springtime and buds, sensuality and fecundity. Wood needs moisture to thrive.

In Chinese medicine, wood is associated with negative feelings of anger, positive feelings of patience, and altruism.