Just a bit more info I left out. Direction is due south so sun all day - roofline is fairly close - less than 10 ft away. Provides shade for bay windows directly north and the main spread of the lawn where kids play moving east. Neighbor's driveway is less than 10 ft to the east so don't a grower that is super wide/huge at top that covers her property (and would need tons of trimming if she complains!)
Mulberry was victim of powdery mildew disease 5 years ago but came back fine. Would like good sturdy grower with strong trunk.

A tree that provides shade is very different from a tree that grows in shade. And it's a bit tricky to determine from your post exactly which of these you are looking for......I suspect the former :-)

If looking for a shade-producing tree, try posting this question - clarified as to primary need - in the Trees forum for a much broader range of suggestions. And it helps if you could provide a region/location - zone 10 SoCal is quite different from zone 10 Florida.

FWIW, never heard of a "carolwood" tree and a Google search turns up nothing under that common name. Any more details on that suggestion?

It would be very helpful if you would say where you are. Trees that are appropriate for a harsh, cold climate with snow and ice for several months of the year might not survive where you are. I'm in zone 9 and never heard of a "carolwood" tree, either.

Thanks for the responses. It says at the top of my page I am in Zone 10B - San Fernando Valley area of Los Angeles County. I also noted in my follow-up post that the tree is in full sun facing south, so this is not a post asking for a type of tree to plant in shade. It's for a tree to provide shade. The tree recommended to me by a local arborist was a "Carrotwood" (my bad on spelling) Hope that clears things up.

You posted on the Shade Gardening forum, hence the confusion......this forum is usually frequented by posters discussing what to plant IN shade, not what creates it :-)

Look at the attached link for some good and site appropriate suggestions, including the carrotwood. FWIW, this species is considered invasive in other parts of the country, as is the silk tree or Albizia. 'Fast growing' is always relative - it depends on growing conditions and aftercare as much as on species but in general, trees that grow fast grow large and tend to be prone to weak wood problems. Fast growing is not always the best choice.

Here is a link that might be useful: shade trees for SoCal

Thanks. Found that site googling today and it's excellent. Ironic, but they recommend what I already have - a fruitless mulberry as fastest growing tree to provide shade for a lawn placement with limited lateral space.

You will want to think carefully about ultimate size and also about litter since it's so close to the neighbor's yard. I would NOT recommend Carrotwood. My community planted them as parkway trees and for weeks they drop an awful black and orange seed mess which sticks in shoes and has sharp edges.

Why don't you look around your neighborhood, parks, and city areas to see what trees you like. Then research more to be sure it will work for you. Your city might have some recommendations too. I'm concerned about the tree being 10' from your roofline. Trees usually need more than 10' to spread out.

We just got two new trees--a magnolia and a crepe myrtle, but I think they would be too large for your area.

I just noticed the sunset western garden book has a section on patio trees and also small to medium garden trees.

Crepe myrtle would probably work for that area. One of the taller tree types such as Natchez, or Pink Velour, for instance.

Ironically enough, the city planted crepe myrtle as the main tree in our neighborhood decades ago, so every house has a crepe myrtle out near the street. Ours was removed by the city in 2001 for disease issues. They love the sun and are gorgeous for those brief summer months (weeks?) when they are in full bloom, but I want something with full green leaf pattern and more substantial trunks/branches if not evergreen.
I didn't know carrotwood had nasty sharp droppings! Thanks for the tip given all the kids that play on our law. A few others that I am considering that I would appreciate feedback on if anyone has experience are:

Willow Peppermint Myrtle (Agnosis Fletuosa)
Camphor (Camphor Officinarm)
Palo Alto Sweetgum