The old theory was to remove tree branches with a concave cut, now more is known about the healing process a slanted cut just above the collar where the trunk and branch meet is recommended -- although this does leave a bump some find ugly. The University of Minnesota has a great article on how and why to prune trees and shrubs.
Despite the cold I spent some time underneath the old Japanese Maple outside our living room, breaking off the obviously dead twigs which are easy to distinguish without the leaves. It’s been pruned well over the years and once the dead was removed, it opened the whole structure and will allow for good air circulation when leafed out.
My fingers were too cold to do much more, although a quick walk around the garden revealed that there was much work to be done.