I can only get a couple of them...

Trees Quiz

1. The virgins saw him dying there, but only after searching in the forest.

2. Seems to be sad, but Shakespeare has him getting laid in it.

--weeping willow?

3. Best to plant it where the Rue once grew, but don't lean on it.

4. Its leaves fall to signal autumn, though seldom now in England.

5. She's bonny, and she's home to owls.

--Oak? (Bonny Portmore)

6. Setting for the daughter of a prominent London politician.

7. A small box of this travelling at half past seven, and the dove sailing between.

8. Spotted while travelling in Oxfordshire, but let's have the fruit in capfuls and bushel bags.

--apple?

9. Besets the road to heaven, but hunger is sharper.

10. We go round but dew files over

--you mean flies over? Mulberry Bush?