Question: Do both Coprosma and Dicksonia belong to a genus of tree ferns in the order of Cyatheales?

Context articles: Cyatheaceae
Dicksonia
Cyathea
Cyathea arborea
Cyatheales
Dicksoniaceae
Coprosma
Lophosoria quadripinnata
Masafuera rayadito
Tree fern

The Cyatheaceae are the scaly tree fern family and include the world's tallest tree ferns, which reach heights up to 20 m.
 They are also very ancient plants, appearing in the fossil record in the late Jurassic, though the modern genera likely appeared in the Cenozoic.
 Cyatheaceae are the largest family of tree ferns, including about 500 species.
 Cyatheaceae and Dicksoniaceae, together with Metaxyaceae and Cibotiaceae, are a monophyletic group and constitute the "core tree ferns".
 Cyatheaceae are leptosporangiate ferns, the most familiar group of monilophytes.
Dicksonia is a genus of tree ferns in the order Cyatheales.
 It is regarded as related to "Cyathea", but is considered more primitive, dating back at least to the Jurassic and Cretaceous periods.
 The fossil record includes stems, pinnules, and spores.
Cyathea is a genus of tree ferns, the type genus of the fern order Cyatheales.
Cyathea arborea (Vernacular English: West Indian treefern Vernacular Spanish: "helecho gigante" or "palo camarón") is a plant of the Cyatheaceae family in the order of the Cyatheales.
 Tree ferns are an ancient form of plant life that still survive in tropical forests.
 This species of tree fern is native to the Caribbean, including Cuba, and the El Yunque National Forest in Puerto Rico.
The order Cyatheales, which includes the tree ferns, is a taxonomic division of the fern class, Polypodiopsida.
 No clear morphological features characterize all of the Cyatheales, but DNA sequence data indicate the order is monophyletic.
 Some species in the Cyatheales have tree-like growth forms, but others have rhizomes.
The Dicksoniaceae is a family of tropical, subtropical and warm temperate ferns. Most of the 5-6 genera in the family are terrestrial ferns or have very short trunks compared to tree ferns of the family Cyatheaceae.
 However, some of the larger species can reach several metres in height.
 A number of others are epiphytes.
 They are found mostly in tropical regions in the Southern Hemisphere, as far south as southern New Zealand.
 Larger tree ferns in the genus "Cibotium" were formerly included in Dicksoniaceae, but are now segregated as the family Cibotiaceae.
Coprosma is a genus of flowering plants in the Rubiaceae family.
 It is found in New Zealand, Hawaiian Islands, Borneo, Java, New Guinea, islands of the Pacific Ocean to Australia and the Juan Fernández Islands.
Lophosoria quadripinnata is a species of fern that, according to DNA molecular analysis, belongs to the Dicksoniaceae family, where it is placed in the genus "Lophosoria".
 It is found in the Americas spanning from Cuba and Mexico to Chile.
 In Chile it is present in the area between Talca and Aysén including Juan Fernández Islands.
 In Argentina it grows only in the humid valleys of western Neuquén and Río Negro Province.
 Diamondleaf fern is a common name.
 In Spanish it is known as 'ampe' (from the Mapudungun añpe) or palmilla, but one has to remember that there are several species of ferns called "palmillas" that have larger or smaller fronds, and which grow in colder climates.
 It is a medium-sized plant, growing to about 4–5 feet (though 10–12 feet in a sheltered place at Arduaine Garden in Argyll, Scotland) and even though the rhizome does not grow a trunk, it is clearly related to the other tree ferns due to features that were apparently already present in their common ancestor, like 'pneumathodes', and the rhizome which changed from the dorsiventral symmetry typical of the other ferns, to a radial symmetry typical of tree ferns.
 Their large and multiple pinnate fronds, with the petiole raised adaxially, and the hairs on the rhizome and lower part of the petioles, also resemble those of tree ferns.
 To identify the species, use the position and characteristics of the spores found on the fertile fronds.
 The genus already existed in the Cretaceous Period in southern Gondwana according to fossil remains found in Antarctica.
 The species is well known as an ornamental plant.
The Masafuera rayadito ("Aphrastura masafuerae") is a rare bird endemic to Alejandro Selkirk Island ("Isla Más Afuera", Outermost Island) in the Juan Fernández Islands.
 The species is a member of the ovenbird family and only one of two species in the rayadito genus.
 The species' natural habitat is humid montane scrub, dominated by tree ferns ("Dicksonia externa") and ferns ("Lophosoria quadripinnata") between 800–1300 m above sea level (though they will descend lower during the austral winter).
The tree ferns are the ferns that grow with a trunk elevating the fronds above ground level.
 Most tree ferns are members of the "core tree ferns", belonging to the families Dicksoniaceae, Metaxyaceae, and Cibotiaceae in the order Cyatheales.
 In addition to those families, many ferns in other groups may be considered tree ferns, such as several ferns in the family Osmundaceae, which can achieve short trunks under a metre tall, and particularly ferns in the genus "Cibotium", which can grow ten metres tall.
 Fern species with short trunks in the genera "Blechnum, Calochleana, Cnemedaria, Culcita" (Europe's only tree fern), "Cystodium, Leptopteris, Lophosoria, Sadleria, Thyrsopteris" and "Todea" could also be considered tree ferns in a liberal interpretation of the term.