The problem, and what I've tried so far
From the screen shot you can see that sky-blue from the mist color is leaking through the trees due to white spots that are occurring in the mist pass on the tree particles. I have no idea what is causing these anomalous white patches in the mist pass.
I created this very simple file to illustrate the problem, but the tree itself was pasted in from a larger project and was created using the Sapling add-on then converted to a mesh for particle use. I gave it a new and very basic leaf material, but the problem persisted, so I believe it is not a problem originating with the material. I tried increasing the number of bounces, thinking it was an issue with the transparency of all the leaf planes, but that didn't help either. I tried flipping the normals of the leaf planes, as well as checking "Double Sided" in the Mesh Context, but it made no difference. To clarify, I am using Cycles.
A similar issue
I found another question here that at first looked to be the same, but that person was using Blender Internal, and using hair particles instead of object particles. The solution is their case was specific to Blender Internal, it seems.
When I tried reproducing the problem using simple cube particles I could not. It appears there is something different about these trees. It's quite puzzling. And notice that some parts of the tree have the correct mist depth while others have the problem. Also note that the problem does not show up in viewport rendering, only in the final render. It's not specific to particles either.