If I may be so bold as to elaborate on what Hoppy was saying...
(TL;DR - The carbon in charcoal and most plant material is very stable. It doesn't dissolve easily, and is not a useful carbon source to your plants.)
I'm pretty sure that the carbon in your standard carbon filter is not available to plants - here's why: Carbon makes a lot of different bonds in a lot of different compounds, and the reactivity and biological availability of those compounds varies as well. Diamonds are an extreme example of an unreactive, highly stable carbon compound. If you dump diamonds in a tank nothing will happen. That carbon doesn't leach into the water - it stays with the diamond. Same goes for graphite - it's made of very stable sheets of pure carbon. And the same goes for the activated carbon in a filter. As with other pure carbon compounds, all the carbon-carbon bonds are very stable and don't break down or bond with other compounds easily - in fact, you would have to set the charcoal on fire to make this happen. For this reason the carbon doesn't leach into the water in any biologically usable form. It does nothing to provide carbon for plants.
I would speculate that most of the carbon in plant matter isn't very useful either. Remember, plants already spent a bunch of energy make those carbon compounds, and some of them are extremely stable as well. Cellulose for example, won't dissolve into any usable form no matter how long you soak it. Now there are special organisms that can digest these compounds and expire them as usable CO2, but I don't think this process is fast enough in a normal fish tank to make any substantial difference in CO2 to the plants. On the other hand, the slow decomposition of, say, a handful of leaves will probably leach other nutrients into the water, so it may still help plant growth. The benefit just would just be comparable to adding a fertilizer, not dosing CO2.
The plural of 'anecdote' is 'data.'