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I found 2 of these (see photo) on my 10-yr-old trident maple. They were on the trunk up near the branches. The tree leafed out beautifully in early May and suddenly most of the leaves dried up and turned brown. Each critter is dome-shaped and is about 3/16" across. Their outer covering is somewhat soft and there seemed to be some very fine sawdust at the point of attachment. What are these things and do I need to worry?

I would have said scale insect, but in whatever country you are in there could be other insects. If as you say the leaves have changed I would have thought that they may be the cause, of course there could be other reasons, like did the tree become dry at some point? Scale insects can be treated with Provado or similar systemic insecticide, this will kill the insect and then when the eggs hatch kill the next lot. It is effective for about two months, however it will also kill any other insect that comes onto the tree.
You could also use methylated spirit on a cotton but which will kill them but you will need to keep doing it,

Thank you for your response. I live in the western USA. The climate is called "high mountain desert". Elevation is 4,750 ft. (1,500 m) and humidity averages 10% - 15%. I will research your suggestions further. Thank you.