Answer: Unfortunately, based on my experience I would not really recommend any of these for use under a street tree. Delosperma requires perfect drainage, such as a gravelly soil, so I don't think it would be such a good choice for this purpose. The creeping thyme tends to die out in patches when used as a ground cover; it also needs a well drained spot (preferably sandy soil) with full direct sun all day to do well. There are two phloxes that might be used as a ground cover, Phlox subulata and Phlox stolonifera. P. subulata requires full sun all day long to grow well, it is somewhat evergreen so it might make a nice groundcover if the spot is very sunny as it would be under a young, newly planted tree; it does not however fill in quickly. P. subulata needs a woodsy location with dappled light or morning only sun and a very humusy organic soil, so it is not well suited for use under a street tree.
If I were selecting a groundcover to go around a street tree, I would look at Vinca minor (glossy evergreen foliage, lovely blue flowers in the spring) or one of the Ajugas (semi-evergreen, some with colorful foliage, blue flowers in the spring) and possibly also liriope if the spot is fairly shady already. These are all very sturdy, reliable plants capable of surviving under harsh conditions. The first two can be planted beneath a new tree and will be able to adapt as the area becomes shadier over time.
Finally, I should mention that it is usually better for a newly planted tree to be allowed to root unimpeded by competing plants. A three inch deep layer of organic mulch applied in a flat layer over the root area year round (without touching the trunk) is the ideal treatment. Rake and replenish the mulch periodically to maintain it at the desired depth and prevent weeds growing in it.
Your local professional nursery staff may also have suggestions based on a more detailed understanding of the growing conditions at the planting site and your personal design goals. Good luck with your tree!
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