Mr.fixit,
May I ask WHICH ball you no longer have in the tray?
If you do not have the “i” ball, right click on the “a” ball and see if you have “Merge into Avast icon” checked.
If so, uncheck it. Avast can merge the two icons into one to clear up the tray.
I suspect this may very well be the problem. Sometimes it’s the simplest things that vex us the most, “Techie101”. 
IF the “a” icon is missing, then try to activate Avast from the Start menu folder.
If you know how, open up the Start.ini file using the MSCONFIG utility. Do you see the file Ashdisp.exe?
Oooooh, I wouldn’t go fooling around with your .ini file…or your .outie for that matter, or even “Miss Config.” Your 'puter might not be too thrilled given you are “Tech101”…I’d wait until I was at least at Tech403 level…But’s that’s just me. ;D
Really, try the first suggestion. Highly unusual for a programme to disappear from the taskbar, next to the clock. Tad suspicious.