hi! the ones that have two forms, one ending in an -en, one not, are both valid; the -en is called a 'singulative' ending: so 'bedw' is birch trees in general, but a single specific birch is a 'bedwen'. Ditto 'gwern, gwernen' etc.
birch bedwen, bedw
rowan cerdinen, criafol
Both fine. You may also find 'cerddinen'.
'gwernen' is a single alder tree, the alder in general is 'gwern' and is directly cognate with Irish fearn, 'alder'.
willow helygen, helgen
Or 'onn' for ash trees in general, esp in older Welsh.
hawthorn draenen wen, ysbyddaden
Again both fine, but Ysbyddaden is probably a bit archaic. (Cf. the name of Olwen's father...!) 'draenen wen' simply means 'a whitethorn'
oak derwen, dar
Both fine, but see above for the singulative. 'Derw' is also possible.
holly celynnen, celyn
or 'coll'...again as above.
But a single aple tree is 'afallen'
ivy eiddew, iorwg
broom/fern, reed eithin/rhedynen, cawnen
you can also say 'corsen'. Again with the -en ending.
blackthorn draenen ddu, draenenwen
Some confusion here. Draenen ddu is 'blackthorn', but draenenwen is 'whitethorn' usually meaning 'hawthorn' - indeed as you give above.
elder ysgawen, ysgaw
Both fine. The second is a partial borrowing of Eng. 'pine'.
aspen(white poplar) aethnen
You can also say 'gwyddwydd' for the latter.
gooseberry eirin Mair
'The Virgin Mary's plums'!