Interesting. I always thought the common misconception was that the tree was thought to be a reference to the New Forest, whereas the original intent was a nod to the various city parks. From those notes, it appears it’s actually supposed to be growth and strength, but it also actually is the NF.
Edit; yep, just checked back a few pages on this thread and this post from Fan the Flames (which I admit also rang a bell with me when he originally posted it).
There was a thread on the badge on the message board before this one (what was that called again) and Rolland Parris' son/ grandson came on and said the tree represented the green parks of the city. Growing up outside of Southampton I had always assumed that the tree represented The Dell, not being aware that most believed to be the for the New Forest.