I took your first tree and did an edit. I liked the basic placement you started with, but it needed a lot of work. I'm not that good at pixel art yet, but I think I can help.
Problem 1: Consistent lighting.
I added a sun to keep track of where the light is coming from.
Problem 2: Black outlines.
I replaced the black outlines around the leaves with the darkest green (as a placeholder). Then I started to work on the trunk, using the lightest brown where the sun hits (up and left), and the darkest brown where the shadow is (down and right).
Problem 3: Flat branches
The trunk and branches all have the same flatness to them. On a real tree some branches would come towards the viewer and some away from the viewer. Branches that come towards the viewer would show a lot of the underside of the branch, so there should be a large shadow underneath. Branches going away would show a lot of the top of the branch, so more of the light brown color.
Problem 4: Not enough branches.
What you have looks ok for Charlie Brown's Xmas tree, but most trees would have additional branches. I added a few smaller branches and tried to add depth too. Some should come towards the viewer, some away. It's harder to do with smaller branches because there are fewer pixels, and some of these may get covered up by the leaves, but it's worth doign well.
Problem 5: Trunk and branch roots too thin
Most trees are like long cones. Wide at the bottom and narrower at the top. The branches work the same way. There are exceptions (palm trees), but this looked like a normal tree. This was an easy fix, just widen the bases of the trunk and branches.
Step 2, the leaves.
Problem 1: Not enough leaves
Too many gaps in the foliage, and the new smaller branches needed some green too. I connected the clumps you had.
Problem 2: Inconsistent shading
With the light from above and a little to the left, the different clumps needed better shading. The leaves also come forward towards the viewer like a dome shape (sorta), so the brighter colors needed to match the dome curve.
This is ok so far, but there's room to improve.
Step 3: Good enough version
Problem: Shading incomplete
Shaded the rest. This was just a lot of filling in, trying to keep the 3d-ness in my head.
Problem: The dark green outline doesn't look so good.
I tried lightening the outline, and then just removed most of it.
Step 4: Final version
Mostly more tweaking of the green leaves on the right hand side. I'm not sure how much of an improvement it is. There's more work that can be done, like adding some leaves in front of the branches, but I called it good enough for an edit. Looks a bit puffball-ish, I suppose.
I stepped away for a bit and thought about explaining what I did, and realized the simplest explanation was the two small circles you see in the last pic (brown and green, just under the sun). Zoom in on those.
Imagine each of these is a sphere rather than a circle. I took the colors in the pic and put them on the sphere where they would show up given the lighting. When you look at a pixel on the tree, you can figure out the direction that surface would be facing and use a pixel of the right color. With only 3-4 colors, the sphere can be small. More colors would probably need a larger sphere, and the ones I drew don't take into account cast shadows. Since the parts of the tree are larger than the sphere (like the trunk), where you choose to fill in the different colors defines the shape of the resulting object.
Things not done:
On this piece the shades were all the same hue. I didn't adjust the colors, but shifting the hues would be a good improvement too.
Hope this helps,