Hm, how big are these trees? How old? Did it have any flowers early in the spring? Did they put on a nice flush of new leaves during the early summer? Could you post pictures of the whole tree(s)?
Mangoes do form flower panicles on the branch tips, and if the trees are too immature to bear fruit, they might abort the blooms, which would look like that. I think a mango tree needs to be at least 4 or 5 years old to start bearing fruit. Maybe more, depends upon the variety.
Most mangoes you'd buy at a store are hybrids that come from grafted trees. A seed from one of those fruits will eventually make a tree that bears fruit, but it may or may not be the type that the seed came from. It also may not have the attributes of the parent tree, like disease resistance and cold tolerance for instance. So when you grow from a seed, you risk growing a nice little tree for 4 or 5 years only to find that it doesn't have good tasting fruit, or it's not cold tolerant and the blooms freeze so you never get any fruit.
But you definitely could luck out and get a tree that bears great fruit. The odds are way better if you start with a grafted tree from a good nursery, though. Pretty much a sure thing.
Are you fertilizing them with a citrus type fert? Making sure they get enough water in dry weather? You're sure it's not some well-intentioned garden helper that has pruned your trees?
Don't know why the tops are dying on yours, if they didn't bloom, but I have baby trees like yours under my big mango tree (from seeds the squirrels left) and none of them have lost their tops like yours. It's a puzzle. Maybe a bug ate the growing tip when it was small and tender.
I wouldn't cut any of the others. Young fruit trees like these will go straight up for a couple of years until they make a sturdy trunk then they'll branch. Wait patiently, you've got a really long way to go!
But, if you really do want fruit in your lifetime, go buy a good variety at a nursery (not Home Depot or Lowe's). Mine was a 3yr. old when I got it, then had one fruit the second year after planting, about 9 the third year and increased production gradually until now (9 years later) we get a really good crop each summer - if we can beat out the squirrels and raccoons!
There's some grasshoppers that will eat the tips (and flowers) of mangos. Also a Mango Tip Borer which is a moth, the caterpillar is what does the damage. There may also be some other local insects in your area. They're usually active at night so seeing you only have a couple of small plants it's easy enough to go out with a light and catch them. Grasshoppers won't move when dazzled by the light and are easy to pick off. Caterpillars are easier still. The alternative is spraying with insecticide but you're better off using the physical method. However, if the growing tips are already gone the insects may have also moved on so you won't see them. But it pays to keep checking especially when new growth starts on the plants again.
I'm just growing these for fun, don't really expect fruit, certainly not with the squirrels around. Thanks for the help, everyone. The tips seemed to turn black and die, so I'm not really sure what's happening with them.