birch vs birchen

birch is a noun but birchen is not a noun.

Word	Noun	Adjective	Verb	Adverb
birch	Yes	Yes	Yes	No
birchen	No	Yes	No	No

As adjectives, birch and birchen are synonyms defined as:

Other synonyms of birch include birken.

birch (noun)	birchen (noun)
a switch consisting of a twig or a bundle of twigs from a birch tree; used to hit people as punishment
any betulaceous tree or shrub of the genus Betula having a thin peeling bark
hard close-grained wood of any of various birch trees; used especially in furniture and interior finishes and plywood

birch (adjective)	birchen (adjective)
consisting of or made of wood of the birch tree	consisting of or made of wood of the birch tree

birch (verb)	birchen (verb)
whip with a birch twig