I needed to lop a couple of branches from a mature silver birch about
10 days ago. The branches are about 2 inches diameter, 6 feet from the
ground and were cut in the normal place just out from the trunk. Since
then the cut part has been dripping sap at a rate of about 2 drips a
second and is showing no signs of stopping or slowing.

Update, had an idea and went out with a roll of cling film. It appears
to have reduced the flow significantly (although not completly stopped
it) hopefully it might give the tree more of a chance to heal.

The cutting this time of year wasn't by choice, had to be done to fit
a new fence, but a branch lopped off 6 weeks ago "healed" fine and
hardly dripped at all so I hope the fact it's pouring out now may be
partly related to the barmy weather in the last 10 days.

David C