You are right, there should have been two pictures. First, here is a photo of the entire tree related to the picture I posted above:

Here's a closer look:

Next, here is the little tree that I described above, but the photo didn't show. I'm wondering what I should do with the shoots here:

I actually have another question about a third tree in my backyard - Here's a picture of it:

Im finding all these little things coming up within a four foot radius of that tree. I wonder if they are offshoots of that tree and whether I should pull them. (sorry for the sideways picture)
Thanks!

The first tree looks more like a Pagoda dogwood to me, not a Japanese maple. :) And yes, you should trim off any "suckers" or shoots that come up from the base of the tree.

For the large green shoots in the smaller tree (with the red leaves), yes-- trim those off as well, and keep trimming anything that comes up. Those are coming up from below the graft, so they'd be a large upright green Japanese maple (the root stock), not something that would match your tree.

As for the little seedlings under the large red Japanese maple, you could probably just pull them up. Are they seedlings?

Thanks magpie! I'm not sure what a seedling is, but I'll go with it!

I just wonder if the little ones growing out of the ground are from the maple's seeds that fell last year. I'd weed them and pull them out. As a general rule, if something sprouts from the trunk or down low on the plant, get rid of it! ;) Allowing them to grow ruins the form of the tree and saps energy.

I started pulling up the little ones - they come out really easily and I bet you are right - they are probably from the seeds last year. Thanks!