I tried a quick Google search to see if this is true, and they don't seem to agree. (However, they are mostly talking about individual trees, not whole forests.) A couple links:
1. "Traditionally, two conflicting viewpoints about tree growth have coexisted. One view was that after an initial increase in growth rate, trees would grow more slowly as they age. Another theory posits that growth rates should increase with increasing tree size. The study results support the latter, showing that in most cases growth rates increase as trees age. This was found to be true for 87 percent of the species reviewed. For the group of largest trees in the study, scientists observed this phenomenon in 97 percent of the species. This translates into a growth rate nearly three times faster for trees with a trunk diameter of 100 centimeters (39 inches) than for trees with a 50-centimeter (20-inch) diameter within a species."
http://environment.yale.edu/yer/article/carbon-capture-tree-...
2. "A 2,000-year-old giant sequoia is just cranking out wood, said Steve Sillett, a professor at Humboldt State University in California who has conducted recent research on the big trees. Other long-lived trees like coast redwoods and Australia's Eucalyptus regnans also show an increase in wood production during old age, according to an article Sillett published in the journal Forest Ecology and Management."
http://news.nationalgeographic.com/news/2012/12/121205-sequo...
3. "This supposition is supported by the fact that we, like others (15), see no consistent evidence for changes in growth rates for individual trees during their lifetimes"
The above papers seem to be an attempt to answer the question, "Can't we just re-capture the carbon released from deforestation/logging by re-planting small trees in these areas," and the answer seems to be "not on a 1:1 basis".