None, though I seem to remember reading somewhere that the group of barrows adjacent to the Sanctuary are placed just below the ridgeline for maximum optical effect (which would have been meaningless had they then been covered with trees :-)

Also, though from a later age -

"Then the Storm-Geat nation constructed for him
a stronghold on the headland, so high and broad
that seafarers might see it from afar.
The beacon to that battle-reckless man
they made in ten days...

Then the warriors rode around the barrow,
twelve of them in all..."*

Again meaningless if the plan was to cover the barrow with trees. It's worth visiting Sutton Hoo by the way as the above almost describes the Ship Barrow there, the headland and the river winding to the sea below (and it's not that far East ;-)

Wot this thread boils down to (or should boil down to) however is preservation. Trees on barrows aren't helping to preserve them any more than oak trees growing three foot from your kitchen window are helping to keep your drains and foundations in good nick :-)

* Beowulf. Translated by Michael Alexander. ISBN 0-14-044268-5. Page 151.