English plum trees, like other fruit trees, must be pruned at certain times of year to avoid hurting fruit production if they are pruned at all. They basically only need to be trimmed to remove damaged limbs and disease. However, you might prune them to increase air circulation or keep height low for ease of harvest. English plum trees generally retain a nice shape, and unless there is a rogue branch that grows unnaturally for some unexplained reason, there is no need to prune them for shaping purposes, except when height is an issue.
Cut damaged limbs off near the trunk. Leave a small nub of a half inch to an inch out from the tree. Cutting a little ways out from the tree will keep you from sawing into the main trunk. Use loppers if the branch is thin.
Prune out diseased limbs several inches past the disease in an area of healthy wood. This will ensure that you get it all. Cut at a notch in the tree limb where another limb is branching off or a clump of leaves is growing.
Prune the top of the tree once it reaches the desired height. Wait for a fork in the top of the tree to form. Always keep the tree trimmed to this height or at the least cut out one side of the fork to keep the tree growing straight and reduce weakness.
Make pruning cuts in summer, not winter. Plum trees are susceptible to silver leaf disease that can develop in cut areas. They are less likely to get the disease in the summer when they are not under stress from the weather and going dormant.
Trim back to the first healthy bud in the second year when pruning for shape. Only prune new growth to adjust the shape of the tree. Older branches are already grown and shaped at this point and only need cutting in the case of disease or damage. Cutting the older branches will affect fruit production.
Cut out branches that are crossing over others and rubbing the bark. This could result in openings for disease.
Select a few branches spread throughout the tree and remove them to allow light into the center of the tree. This will help with fruit production.