Okay thanks! I know in Japan they call them Sugi, but I wasn't sure if we called them the same.

I want to thank both benz6 and Embothrium for taking the time to answer and getting pretty close to the description of what I have seen of the tree in question. Yesterday, although in evening light, I was able to get another look at it. I couldn't get close to any of the roundish orange clusters still on the tree, but after checking some of the pictures online, I feel pretty sure it must be a carrotwood. The Pacific madrone, in most of the pictures, appeared to have a red bark, and in some pictures it was shown peeling, whereas the tree I was searching for definitely has a smooth grey trunk. (Though, interestingly, one source I found through Embothrium's link mentions that the carrotwood has an orangey or red bark under that smooth grey one.) In one of the pictures, I could also see black seeds peeking out at the end of their orange sheaths, right on the tree, just like the ones the little girl showed me. And those seed capsules, or whatever they are called, are slightly elongated ovals--again, like the ones she brought to me. I'll have a chance for a look at the tree in better light on Sunday, but am satisfied for now that the mystery has been solved. Interestingly, a YouTube video in the linked materials says that this tree is invasive and not welcome in some places, but in Florida and California, it is, nevertheless, cultivated for ornamental purposes. That would accord with my finding more than one specimen planted along a busy street in San Diego. Must check if there are many, next time I am in that area.

Now that we have identified this as a Beauty Leaf Tree (Calophyllum), I have a followup question. I have one of these trees outside my front gate. But beyond the back line of my property, between the edge of pavement and my wood fence, I have this tree all over. This is an area about 40' long and 18' wide, and it's completely crowded with this tree. I did some counting today, and within this 40'x18' area, I have three large ones where the trunks are over 18" in diameter. I have about fifteen with trunks between 8" and 10", and I have numerous (several dozens) between 4"-8". Altogether I think I have over forty beauty leaf trees in various sizes growing in this small area. It's so thick and dense I cannot get inside this cluster of trees. It drops A LOT of fruits/seeds around this time of the year and previously owners just let them go crazy. I am trying to decide if I should cut back the small trees and over time get back to just the three large mature ones, or leave it alone and let it be? Thoughts?

Thanks for all contributions. The second tree one poster said could be a walnut has never produced nuts of any kind. And the thing is, I would like to remove the ash by the house, and I do worry about it getting too large both height and root wise, but I also don't like nagging and that is what I'd have to do to my husband to get rid of it. He likes this whole "natural" look in the yard; the first tree has a wild grapevine growing through it as if it's a trellis.