This is the first time I have been asked about this.
Most builders prefer to cut down most of the trees anywhere near proposed buildings before making a planning application if they are not already protected.
No one can do anything to a tree with a Tree Preservation Order on it without the TPO officer's prior approval and agreement. You can also be sure that in "austerity Britain" the council will make a charge for the before and after site visit and any authorisation approval.
How to make an application to carry out work on a TPO tree.I am surprised that Countryside didn't choose to tidy up the tree so it looked better and therefore the home was more likely to sell, perhaps for a higher price.
I would get a quote for the tree surgery and suggest that you ask Countryside to pay half the cost would be fair.
I will add the subject of Tree Preservation Orders to my main website on the
"Questions to Ask" page!