Sounds like winter moth, although not sure they're small enough to fit your description (not sure what 1/16th of an inch looks like in mm, seems more like egg size), but I can think of no other pest which causes this trouble at that time of year. The way to handle that is to apply a greaseband or barrier glue to the tree's trunk in October, to prevent the wingless females from crawling up the trunk to lay their eggs in the crown of the tree, and ensuring nothing else sticks to the band between then and mid April, in other words, that it remains sticky or greasy.
Obviously too late to protect it this year, but you may find it's not so much of a problem this time round - if there is a heavy infestation, you can spray shortly after bud burst, but not while the flowers are present (so that pollinating insects are not killed). Sprays available are Bayer sprayday Greenfly Killer, or for a more organic treatment, Py Spray Garden Insect Killer, based on pyrethrum.