"If enough of us register our approval/disapproval, it would send a very powerful message to The National Trust and to all owners of heritage trees, that they should think very carefully about the way they manage these unique and irreplaceable green assets," he said.
The 190-year-old tree, which stood at the trust's Kingston Lacy property in Dorset, was felled in December.
Barrell, who first examined the tree shortly after the National Trust acquired the property in 1981, has said that indications before and after its felling show it to have been in reasonable health.
The trust has said that the proposed alternative of branch pruning would have been problematic and would have left the tree aesthetically unappealing.
The video, recorded at the property during the tree's felling, has so far gained over 2,500 hits and received 240 "likes".