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Adventitious has various meanings in various disciplines and in general usage.
Adventitious is from the Latin root advenire, meaning "to come or be superadded" and in correct English the meanings tend to have connections to accidental or casual occurrence. "...of the nature of an addition from without; supervenient, accidental, casual.
Buds and shoots
Adventitious buds develop from places other than a shoot apical meristem, which occurs at the tip of a stem. They may develop on stems, roots or leaves. Shoot apical meristems produce one or more axillary or lateral buds at each node. When stems produce considerable secondary growth, the axillary buds may be destroyed. Adventitious buds may then develop on stems with secondary growth.
Adventitious buds are often formed after the stem is wounded or pruned. The adventitious buds help to replace lost branches. Adventitious buds and shoots also may develop on mature tree trunks when a shaded trunk is exposed to bright sunlight because surrounding trees are cut down. Redwood (Sequoia sempervirens) trees often develop many adventitious buds on their lower trunks. If the main trunk dies, a new one often sprouts from one of the adventitious buds. Small pieces of redwood trunk are sold as souvenirs termed redwood burls. They are placed in a pan of water, and the adventitious buds sprout to form shoots.
Some plants normally develop adventitious buds on their roots, which can extend quite a distance from the plant. Shoots that develop from adventitious buds on roots are termed suckers. They are a type of natural vegetative reproduction in many species, e.g. many grasses, quaking aspen and Canada thistle. The Pando quaking aspen grew from one trunk to 47,000 trunks via adventitious bud formation on a single root system.
Some leaves develop adventitious buds, which then form adventitious roots, as part of vegetative reproduction; e.g. piggyback plant (Tolmiea menziesii) and mother-of-thousands (Kalanchoe daigremontiana). The adventitious plantlets then drop off the parent plant and develop as separate clones of the parent.
Coppicing is the practice of cutting tree stems to the ground to promote rapid growth of adventitious shoots. It is traditionally used to produce poles, fence material or firewood. It is also practiced for biomass crops grown for fuel, such as poplar or willow.
Adventitious rooting may be a stress-avoidance acclimation for some species, driven by such inputs as hypoxia/anoxia or nutrient deficiency. Another ecologically important function of adventitious rooting is the vegetative propagation of tree species such as Salix and Sequoia in riparian settings.
The ability of plant stems to form adventitious roots is utilised in commercial propagation by cuttings. Understanding of the physiological mechanisms behind adventitious rooting has allowed some progress to be made in improving the rooting of cuttings by the application of synthetic auxins as rooting powders and by the use of selective basal wounding. Further progress will be made in future years by applying research into other regulatory mechanisms to commercial propagation and by the comparative analysis of molecular and ecophysiological control of adventitious rooting in 'hard to root' Vs. 'easy to root' species.
Adventitious roots and buds usually develop near the existing vascular tissues so they can connect to the xylem and phloem. However, the exact location varies greatly. In young stems, adventitious roots often form from parenchyma between the vascular bundles. In stems with secondary growth, adventitious roots often originate in phloem parenchyma near the vascular cambium. In stem cuttings, adventitious roots sometimes also originate in the callus cells that form at the cut surface. Leaf cuttings of the Crassula form adventitious roots in the epidermis.
Tuberous roots are without any definite shape; example: sweet Potato.
Nodulose roots become swollen near the tips; example: turmeric.
Prop roots give mechanical support to the aerial branches. The lateral branches grow vertically downward into the soil and acts as pillars; example: banyan.
Climbing roots these roots arising from nodes attach themselves to some support and climb over it; example: money plant.
Adventitious roots and buds are very important when people propagate plants via cuttings, layering, tissue culture. Plant hormones, termed auxins, are often applied to stem, shoot or leaf cuttings to promote adventitious root formation, e.g. African violet and sedum leaves and shoots of poinsettia and coleus. Propagation via root cuttings requires adventitious bud formation, e.g. in horseradish and apple. In layering, adventitious roots are formed on aerial stems before the stem section is removed to make a new plant. Large houseplants are often propagated by air layering. Adventitious roots and buds must develop in tissue culture propagation of plants.
In law, adventitious can mean "falling to a man by mere fortune, or from a stranger..."
In medicine, adventitious and related words such as adventitia refer in various senses to the basic meanings associated with the Latin root advenire. Typically the general intention is to convey something like: of the nature of a later addition from without, accidental, casual.
On similar principles, adventitious in medical terminology can refer to conditions acquired after birth, particularly if the cause is obscure, casual, or accidental.
In auscultation the term adventitious refers in general to any diagnostic added sound not normally to be expected in the healthy body and likely to indicate an undesirable condition. The terminology is confused, largely for historical reasons, but roughly speaking one listens for crackles and wheezes in the respiratory passages, and rubbings in pleurisy.
A wide variety of other sounds are correspondingly relevant in cardiovascular conditions, fractures, joint problems etc.
Adventitious presence in agriculture refers to the accidental or unintentional appearance of foreign material in a product. This usually happens in the production, harvesting, storage, and marketing of grains, seeds, or food products.
Grain and seed companies argue that virtually all shipments contain some type, and level, of adventitious material, such as weed material. Generally, buyers recognize that some level of adventitious material is acceptable, and foreign material limits are specified in purchase contracts.
Adventitious presence is a key issue in the debate over regulation of biotechnology. As more and more crops, and acres, are devoted to genetically engineered (GE) varieties, it becomes increasingly difficult to segregate these from GE-free varieties, which some buyers and countries demand.
The Dalai Lama describes the autopilot thoughtstreams as adventitious, in a speech about Rigpa awareness: "However, this primordial quality of buddhahood is obscured by adventitious mental factors, our afflictions and other thought processes." 
- Brown, Lesley (1993). The New shorter Oxford English dictionary on historical principles. Oxford [Eng.]: Clarendon. ISBN 0-19-861271-0.
- Drew et al. 1979 Ethylene-promoted adventitious rooting and development of cortical air spaces (aerenchyma) in roots may be adaptive responses to flooding in Zea mays L. Planta 147 1; 83-88), (Visser et al. 1996)
- Naiman and Decamps, 1997, The Ecology of Interfaces: Riparian Zones. Annual Reviews in Ecological Systems
- Klerk et al. 1999 Review the formation of adventitious roots: new concepts, new possibilities. In Vitro Cell & Developmental Biology - Plant 35 3;189-199
- McVeigh, I. 1938. Regeneration in Crassula multicava. American Journal of Botany 25: 7-11. 
- Warrell, D. A.; Weatherall, D. J.; Ledingham, J. G. G. (1996). Oxford textbook of medicine. Oxford [Oxfordshire]: Oxford University Press. ISBN 0-19-262140-8.
- This section This article incorporates public domain material from the Congressional Research Service document "Report for Congress: Agriculture: A Glossary of Terms, Programs, and Laws, 2005 Edition" by Jasper Womach.
- Dalai Lama, J. Cabezon translator, 2009. Meditation On The Nature Of Mind.
- Esau, K. 1977. Anatomy of Seed Plants. New York: Wiley.
- Hartmann, H.T. and Kester, D.E. 1983. Plant Propagation: Principles and Practices. Englewood Cliffs, NJ: Prentice-Hall.