It can take several years of staking and some pruning to get the heavy top growth under control.

Do not tie the trunk snugly to the stake like a splint. It will never get stronger.

Support the trunk between two stakes, with a bit of slack so the trunk can have some movement. I use old pantyhose because they flex and stretch a bit.

The stakes should be on opposite sides of the tree, about a foot away from the trunk.

Place the ties high enough in the tree so the top doesn't flop over, and also one or two ties further down to help support it.

Carefully shorten and prune out the top branches to make a good top structure and remove the excess growth as needed. Mesquites have rampant top growth that is too much for the trunk. I've removed as much as 2/3 the top of a young mesquite that was flopping under the weight.

Every few months, check by loosening the ties and see if it's supporting itself ... eventually it will be able to do so and you can loosen the ties more, and then remove them when it's standing on its own.

Thank you for your advice. My question was addressing what would happen if we chose NOT to stake the tree. If we don't stake the tree, will it grow like those we see around town or will it fail to thrive?

I don't know. Some make it, and others don't develop enough of a root system to survive.

Our wild mesquite trees do fine and don't bend. A potted, staked one is now bending without its support. It seems that they should never be staked.

If I let it flop over, will it eventually straighten itself out, or do I have to keep it staked?