This unprickly prickly pear tree is growing in front yard of a high-rise residence in Waikiki. I've ruled out so many names that now I'm down to no name at all. My latest guess was Consolea moniliformis, based on photos at Consolea moniliformis, showing this actually in a garden in Hawai'i, but the "densely armed with spines" sentence, and a photo by Tony Rodd at Consolea moniliformis 080315-4346 have convinced me that it can't be that. I have two postings-worth of photos. Here is the habit - a tree with pad-like leaves, but not flattened as if run over by a truck, and then flowering bits at the margins of the pads. Here are some photos of the ends of the branches showing some small flowers and buds. The pads do fall off, and new plants seem to take root easily. I have a few more flower photos, have to be in the next posting.
Here are photos of the small flowers, either yellow or orange. It's not obvious that one colour fades to the other. No flowers were red. The largest flowers were only slightly more than 2cm in diameter. There were no fruits. I wanted to add some links to resources that would seem to be really good, but I didn't find my ID from them. On-line Guide to the positive identification of Members of the Cactus Family Introduction to Opuntias: THE Signature Cactus Genus - Dave's Garden Cactaceae
Road Kill Cactus (Consolea rubescens)? Consolea rubescens (Road Kill Cactus) | Cactus, Cacti, succulents, Succulent landscaping
Thanks, @Sulev. I have to say that the photos at On-line Guide to the positive identification of Members of the Cactus Family make this look like a really good match. Photos at PlantFiles Pictures: Consolea Species, Road Kill Cactus, Sour Prickly Pear (Consolea rubescens) by Kell look like a good match too. The first three photos at Photos of Road Kill Cactus (Consolea rubescens) ยท iNaturalist look like mine, but the fourth one, a very fuzzy photo, seems to show spines. What put me off this ID were the photos at Hybrid Epi Cactus Display Page, with the description of the stems as seeming "to have been run over and flattened by a truck" (I thought mine were not flattened enough, as I mentioned in my posting, but maybe they are) and the spines along the pad margins. Lots of spines are also seen in the Smithsonian photos at Plants Profile for Consolea rubescens (sour pricklypear). Also those photos show a red flower, which would seem to match the name. Maybe the Smithsonian ID is wrong? Maybe a few spines along the pad margins at the other site are not uncommon in youngsters? There are no spines to be seen on my photo of the sprouts in the grass, though.
Here is another page with many photos that support this ID, but two photos with spines. Consolea rubescens These two pages say "few or no spines". Succulents-Consolea rubescens Road Kill Cactus (Consolea rubescens) Well, maybe this covers it: "old trunk either naked or spiny": Consolea rubescens | The Cactus King
Those small green pads are 'sort-of' fruit - they are what develops into fruit in normal Opuntia (& related genera). Looks like they don't develop into normal fruit but do develop further budding at their apices, resulting in those chains of 'fruit'. Presumably it only reproduces vegetatively (in this individual, at least).