My go-to camping spot is a walk-in site about 100 yards from where I park, so while not exactly car camping, it allows me to bring extra stuff and dig it out if I want to use it. That usually means a bunch of knives, a folding saw and a hatchet. Last time out, I mainly used a Mora Companion and a folding saw for fire prep, and a Kizer folder for other general cutting tasks. The trip before that, I used a hatchet, a fixed Schrade X-Timer XT2B and a Coast FX350 folder. Variety being the spice of life, and all that.
If the botanical garden you're describing is anything like the state parks here, you're supposed to either buy firewood at the camp office or from a state-certified retailer. I've scavenged a fair amount of fallen branches to supplement purchased firewood when staying multiple nights, as well as twigs for my folding stove, but chopping anything down is strictly forbidden, and fires are too depending on conditions. I would check prior to your trip. Getting back to the knife, though, really anything with a flat or scandi grind is great for most purposes, but I do like having both a folder and fixed blade for convenience.