I am unable to click on a tab within the tree to switch to it. All other functionality of the extension appears to be working.
Clicking a tab highlights it to show it is selected. This causes both the current tab and the clicked tab to show as highlighted.

(while in new issue tab)
The focused tab switches to the tab that is clicked on.
The tab is marked as selected but nothing happens.
I have TST at home with the same Firefox version on Linux, where the issue does not occur.
You really use TST 3.1.2? It is looks too old version.
You really use TST 3.1.2? It is looks too old version.
It looks like the page that opened on install showed the wrong version. I am actually on 3.4.1, updated main post.
I've seen your screenshot: it says you use Multiple Tab Handler 3.1.2 and TST together. I think you use TST 3.4.1, the latest available version on addons.mozilla.org.
There is a known incompatibility #2504 with an addon Move unloaded tabs for Tree Style Tab. The incompatibility has been fixed at latest development version, and a newer released version is still waiting editor's review...
If you don't use Move unloaded tabs for TST, this can be a different issue. But sadly I couldn't reproduce this with Nightly + new profile. There looks to be something more hidden triggers on your environment.
If you don't use Move unloaded tabs for TST
馃う鈥嶁檪 I do. Disabling it has solved the problem.
I did look through open issues a couple of times and completely missed the existing issue. Apologies for that and thank you.
Most helpful comment
I've seen your screenshot: it says you use Multiple Tab Handler 3.1.2 and TST together. I think you use TST 3.4.1, the latest available version on addons.mozilla.org.
There is a known incompatibility #2504 with an addon Move unloaded tabs for Tree Style Tab. The incompatibility has been fixed at latest development version, and a newer released version is still waiting editor's review...