Question: Which type of Elf is mentioned in The Tale of Beren and Luthien?

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Sarcoscypha coccinea: Sarcoscypha coccinea, commonly known as the scarlet elf cup, scarlet elf cap, or the scarlet cup, is a species of fungus in the family Sarcoscyphaceae of the order Pezizales.  The fungus, widely distributed in the Northern Hemisphere, has been found in Africa, Asia, Europe, North and South America, and Australia.  The type species of the genus "Sarcoscypha", "S. coccinea" has been known by many names since its first appearance in the scientific literature in 1772.  Phylogenetic analysis shows the species to be most closely related to other "Sarcoscypha" species that contain numerous small oil droplets in their spores, such as the North Atlantic island species "S. macaronesica".  Due to similar physical appearances and sometimes overlapping distributions, "S. coccinea" has often been confused with "S. occidentalis", "S. austriaca", and "S. dudleyi".

The King of the Cats: The King of the Cats (or The King o' the Cats) is a folk tale from the British Isles.  The earliest known example is found in "Beware the Cat", written by William Baldwin in 1553, though it is related to the first century story of "The Death of Pan".  Other notable versions include one in a letter written by Thomas Lyttelton, 2nd Baron Lyttelton, first published in 1782, M. G. Lewis told the story to Percy Bysshe Shelley in 1816, and a version was adapted by Joseph Jacobs from several sources, including one collected by Charlotte S. Burne.  Walter Scott reported that it was a well known nursery tale in the Scottish Highlands in the eighteenth century.  It can be categorised as a "death of an elf (or cat)" tale: Aarne–Thompson–Uther type 113A, or Christiansen migratory legend type 6070B.

The Dragon (fairy tale): The Dragon (Neapolitan: "Lo dragone" ) is an Italian literary fairy tale, included in Giambattista Basile's "Pentamerone" (Tale IV.5), first published 1635.  In the English language, the tale was a selection in Thomas Keightley's "Fairy Mythology" (1828), and later appeared in John Edward Taylor (fl.  1840-1855)'s translation of the entire work, "The Pentamerone, or, The Story of Stories, Fun for the Little Ones" (1848). The tale has been classed as a version of Aarne–Thompson type 462 "the outcast queens and the ogress queen", rather than as "the dragon-slayer" (type 300).  It exhibits folkore motif K873, "fatal deception by giving narcotic."

Ground dipole: In radio communication, a ground dipole, also referred to as an earth dipole antenna, transmission line antenna, and in technical literature as a horizontal electric dipole (HED), is a huge, specialized type of radio antenna that radiates extremely low frequency (ELF) electromagnetic waves.  It is the only type of transmitting antenna that can radiate practical amounts of power in the frequency range of 3 Hz to 3 kHz, commonly called ELF waves A ground dipole consists of two ground electrodes buried in the earth, separated by tens to hundreds of kilometers, linked by overhead transmission lines to a power plant transmitter located between them.  Alternating current electricity flows in a giant loop between the electrodes through the ground, radiating ELF waves, so the ground is part of the antenna.  To be most effective, ground dipoles must be located over certain types of underground rock formations.  The idea was proposed by U.S. Dept. of Defense physicist Nicholas Christofilos in 1959.

Beren: Beren (also known as Beren Erchamion, "the One-handed", and Beren Camlost, "the Empty-handed") is a fictional character in J. R. R. Tolkien's Middle-earth legendarium.  He appears in "The Silmarillion".  He is a human, and his love for the elf maiden Lúthien is central to the Tolkien legendarium.

The Tale of Beren and Lúthien: The Tale of Beren and Lúthien, told in several works by J. R. R. Tolkien, is the story of the love and adventures of the mortal Man Beren and the immortal Elf-maiden Lúthien.  (Early versions of the story, published in the standalone book in 2017, described Beren as a Noldorin elf.)  Tolkien wrote several versions of their story, the latest in "The Silmarillion", and the tale is also mentioned in "The Lord of the Rings".  The story takes place during the First Age of Middle-earth, about 6500 years before the events of "The Lord of the Rings".

Doriath: In J. R. R. Tolkien's fictional Middle-earth, Doriath is a forest realm of the Sindar in Beleriand ruled by King Thingol and his queen Melian.  It serves as a principal stage for the stories of the First Age, such as The Tale of Beren and Lúthien from "The Lays of Beleriand", and parts of "The Children of Húrin" and "The Silmarillion".  It is called the "Fenced Land" because of a girdle of enchantment Melian put about it, allowing none to enter the kingdom without her leave or Thingol's.

Elf: An elf (plural: "elves") is a type of human-shaped supernatural being in Germanic mythology and folklore.  Reconstructing the early concept of an elf depends almost entirely on texts in Old English or relating to Norse mythology.  Later evidence for elves appears in diverse sources such as medical texts, prayers, ballads, and folktales.

The Lay of Leithian: The Lay of Leithian is an unfinished poem written by J. R. R. Tolkien.  It tells the Tale of Beren and Lúthien, the story of the love of the mortal Man Beren and the immortal Elf maiden Lúthien.  The poem consists of over 4200 verses.  It was published after Tolkien's death in "The Lays of Beleriand".  Its precedents are found in the Welsh tale of Culhwch and Olwen in the manuscripts, the Red Book of Hergest and the earlier White Book of Rhydderch.

The Elf Mound: "The Elf Mound" (Danish: "Elverhøi" ) is a literary fairy tale by Danish poet and author Hans Christian Andersen (1805–1875).  The tale is about a feast held in an elf mound for the Goblin Chief of Norway and his two sons, both of whom are expected to select elf brides.  The tale was published in Copenhagen, Denmark by C.A. Reitzel in April 1845.

Supporting paragraphs: The Tale of Beren and Lúthien Elf
Answer: Beren as a Noldorin elf