A many trees is chiefly dialectal and/or regional. Its form has been in use since at least the late 16th century. On analogy with a few trees it is not "inherently ungrammatical." The Oxford English Dictionary calls the form regional in England. In short, it is grammatical to native English speakers found all over the world; although most native English speakers would not call the form a part of standard English.
Here are some 21st century examples from native speakers from England, Australia, Ireland, and the USA.
It will take a many years and much work to construct a common identity in South Sudan, but the South Sudanese shouldn't be afraid of the challenge.
—James Copnall, 2014, A Poisonous Thorn in Our Hearts, (link) published by Hurst, distributed in North America by Oxford University Press. Mr Copnall (twitter) is a native English speaker and a long-time BBC correspondent.
In Australia we have seen a many natural disasters, including floods, bush fires, and cyclones.
—Dr. Malcolm Freeman, 2015, A Practical Guide - Management of Risks in Small and Medium-Size Enterprises, (link). Dr. Freeman is a native English speaker, who has lived in Australia since 1969 (Facebook page). Natural disaster is, of course, an open compound noun, meaning it is one word; and so a many directly precedes a plural count noun.
BitPim is an open source tool that allows you to view and manipulate files on a many CDMA phones.
-Dr. Darren R. Hayes , 2015, A Practical Guide to Computer Forensics Investigations, published by Pearson Education, USA (link). The sentence appears in the hard copy of the book. Dr. Hayes (Pace University, NY) received his undergraduate education in Ireland, so he might to be a native of that land.
Funkmaster Flex and the Pitbulls DJ Squad are all class acts. We protected them a many nights and they made life easier for us to get through the night (p 224)...My aura has touched a many generations (p 237).
-Ray Mack, 2014, Underestimated: A Searcher's Story, Xlibris (link). He is a native of New York City.
I have limited myself to these four non-fiction examples (although there are plenty more in Google Books) to show how geographically widespread the use of a many + plural count noun is; although yes, the number of speakers who use this form are in the minority, at least from my perspective as a native speaker of American English.
The form has been a part of English since at least the 16th century:
Though a many friends Are made a way.
-Marlowe, Edward III, circa 1593 (cited in OED)
Hee were a mad man that to Secure himselfe from the Fire, would pile a many Billets betweene him and the flame.
-J. Shute, Judgement & Mercy, ca 1645 (cited in OED)
Note well that the forms a good/great many + plural count noun and a good/great many of + plural count noun were also used during this time (along with other adjectives such as considerable/pretty/jolly, per the OED). These forms, using good and great, have prevailed/survived in most people's dialects. But some people can still use the form a many + plural count noun without the intervening adjective.
As for a few (meaning "a small number of") before a plural count noun, we can go back as far as 1297:
Þe kyng with a fewe men hym~self flew at þe laste.
(the king with a few men himself slew at last).
-The Chronicle of Robert of Gloucester (OED)
He shall in a fewe stoundes Lese all his markes.
-Roman Rose, circa 1400 (OED) Note: A stound probably means a moment in this example.
Shakespeare has a lovely line in Cymbeline (1623):
Heere's a few flowers.