Many thanks to those who have responded. To nanw: I think that you nailed the ID of the plant. To greenman28: the subject plant has linear, tapered leaves, and does not look like the usual Kalanchoe leaf. The new leaves on the edge are also different from the usual "Mother of Millions" plants, in that they do not have root structures, and may expand to look almost the same as the original "mother" leaf. To tapla: thanks for the explanation, and the stimulus for me to try to find more about the frequency of occurence of this process in epiphyllum. I did not find any relevant information, so I again turn to forum members. Is this a frequent phenomenon in members of the epiphyllum family? At this moment, I have learned a lot about the family, and have acquired knowledge of the amazing beauty and variety of a plant group I had somehow ignored. I missed the fact that it was a cactus (it does not look like the usual cactus), and thus missed the clue that cactus leaves are "different", and may develop daughter plants from their "leaves". Thanks again for your help!!

Are you giving your nanner full or very nearly full sun outdoors? If its indoors it may just be too dark where you have it. Give it as much sun as possible. That's all that comes to mind.

3 tips have peeled off now. Looks like a 4th will in a few days. I do see signs of life that could turn into a result similar to outdoorgirl's last post. I have not watered at all. Baby leaves on other tips are still falling off at first growth. It's been hot and humid here for a month now with lows in the mid/upper 70s. Her sister is doing amazing with an inflo about 1.5 inches long and healthy leaves. The one black tip she has looks to be containing itself and hardening over. Her sister: this growth tip never turned the dark redish color like the tips on the other plant So frustrating to have two plumies going I'm opposite directions! Maybe it's just time for a third :)

If it were mine and I wanted it pretty much the way it is now, single stem with a bushy top, I would cut it back below the branches on each prong of the 'trident'. Put it outside in partial shade for a couple weeks to acclimate, and then move to full sun. Ficus back bud easily, and full sun will encourage this. Once new growth is under way, move into part shade so as not to produce sun leaves, which will not like being moved into a relatively dark house for winter. Ficus are known to drop their leaves if a shadow passes over them lol. If you want a bushy plant that branches near the soil line, you can cut the main stem to 6-12 inches, and proceed as above. There is a chance, however slight, that the plant will not like being chopped on and turn up it's toes, but I think it's a chance worth taking considering the state the plant is in now.