What I have so far, biomes and their trees after two days of sampling my area:
pinebarren

highground

linchenwold
heath
timberland

wald
sombrebramble

swamp
sandcliff

dirt
beechgrove

flowermeadow

moor

leafpatch

hardsteppe

rootbosk

leaf

bog
redplain

grass
bluesod

cloudrange

dryflat

fen

flowermeadow

greensward

lushfield

mossbrush

oakwilds

shadycopse

snow

wildturf

There are typos; did most of this while half asleep.
I'm quite sure there are missing trees for some of the biomes. Bush list I'm not satisfied with, so I'm leaving out for now.
Will put up on a spreadsheet/wiki some time soon. I'm more interested to learn about the names of other biomes so that I can retrieve and extract all the textures from the server in one go.
I've been using bdew's client to get the name of a biome using the worldtip toggled permanently option.
If you could be so kind as to help this autist fulfill the dream of documenting all the biomes, do help by posting the names of biomes that aren't in the OP.
Will update.
Update
2015/10/03
Clemence wrote:Grassland ( gright green) :
aspen, elm, alder, pine, chestnut, kingsoak, willow,birch,oak,beech,goldenchain,apple tree, pear tree, crabapple, mulberry.
2015/10/03
Added 11 new biomes.
2015/10/04
Added images