The suckers should be removed as they destroy the shape of the tree and
don't produce as much fruit as a lateral branch. You should not remove
more than 1/3 of a tree in any given season. Hopefully, you have enough
lateral branches growing that should be encouraged to grow faster when
the tree re-directs it's energy away from the suckers. Don't do any cutting
while the tree sap is actively flowing, or that will encourage the tree to send
out more suckers. Early Winter or late Fall might be the best time for this
kind of pruning. Try to encourage a single central branch or leader to direct
the tree
to a more pleasing shape, although with your very old tree that may not be
possible.

Sherwin D.

Zootal wrote:

I have a very large very old apple tree in my back yard. It has not been
maintained much the last few years. It consists of a few main branches and
hundreds of suckers growing straight up from the branches. Should I remove
all of the upright growing suckers? I hate to butcher the tree, but it is in
serious need of some TLC.