Your description of the situation sounds like your tree could be Celtis occidentalis (Sugarberry). Pictures of the trunk would help, as would images of any fallen leaves. Pictures are almost always more helpful to us than a verbal description. Let us know your general location, too.

I am planning on having the trimmed down substantially

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first.. positively ID the tree ...

no one here.. approves of topping trees ...

if its a carp tree... get rid of it ... unless you simply want to pay the tree dude to come back every few years.. and take your money ...

heavily trimming an established mature tree.. with no impact on its root mass.. will only result in explosive new growth ...

and one of murphys rules says.. it will grow right back over the driveway .. lol .. trust me on that ...

get rid of it.. and plant a quality tree ... pay once.. and be done with it ...

how about some pix.. its hard to ID a tree in winter.. presumably leafless... but we enjoy trying.. maybe you can find some leaves around.. or a dried berry.. etc ...

geez rhiz.. i was going to say i never heard of this in my z5 MI ... what gives by not calling it a hack berry tree.. lol .. and therein lies the rub with latin and common names ....
hack has a very distinctive bark.. which would help with ID ..

ken

could it be a large mulberry tree? Squirrels here are known to chomp off clumps of leaves with berries on them.

The Bradfords at my work seeded extra heavily this year. Made quite the mess. I beg I can still take pics of mounds of their mess on the sidewalks.

I'm still thinking that it's a hackberry. Gille, your plan of having the tree cut back substantially is not a good one. Ken is right on!

Trees that are substantially pruned, perhaps even (yikes) 'topped', have a predictable reaction just as ken said earlier...they regrow many new shoots rapidly, usually in large numbers of weakly attached branches. What you hope will resolve your problem may result in an even worse mess.

Hackberries are messy, messy trees....pretty undesirable in close proximity to homes, walkways, patios, and driveways.

I hope that you're paying attention to this thread, so that you can help us narrow down the identity of your tree by attaching some images.

Hackberry and Sugarberry have very distinctive bark, so will be easy to id them.