A brief review of recent non-paywalled available literature indicates that such an effect likely exists but that it is difficult to quantify based on currently available data.
Some amount of carbon from trees can be sequestered in the soil for periods time significantly longer than the typical above-ground decomposition time of organic matter, potentially for millennia. This clearly lengthens the carbon cycle time, but it is not clear to me whether this represents carbon storage, as there does not seem to be a well established minimum cut-off time for this. The primary source for soil-sequestered carbon are tree roots, with leaf litter constituting a secondary source.
The following paper (preprint online) addresses the question in the specific context of agroforestry, i.e. cropland interspersed with trees. The paper notes multiple times that the processes involved in soil sequestration are not well understood and that quantitative measurements and estimates vary widely, as one would expect based on differences in climatic and soil condition. Note on units: A Mg corresponds to a metric ton.
Klaus Lorenz and Rattan Lala, "Soil organic carbon sequestration in agroforestry systems. A review." Agronomy for Sustainable Development, Vol. 34, No. 2, April 2014, pp. 443-454.
The soil organic carbon (SOC) pool, in particular, is the only terrestrial pool storing some carbon (C) for millennia which can be deliberately enhanced by agroforestry practices. [...] The SOC storage in agroforestry systems is also uncertain and may amount up to 300 Mg C ha−1 to 1 m depth.
[...] Further, between 30 and 300 Mg C ha−1 may be stored in agroforestry soils up to 1-m depth [...] The SOC sequestration depends primarily on the soil C input and soil stabilization processes. Plant root and rhizosphere inputs, in particular, make a large contribution to SOC [...] Litter fall and in turn SOC sequestration may be affected by stand-density management as, for example, higher stocking levels of trees enhance the vegetation C pool.
Another paper that deals specifically with urban trees mentions in passing that the soil accounts for a significant part of the carbon sequestered in non-urban forests (presumably both naturally growing and human-managed):
David J. Nowak and Daniel E. Crane, "Carbon storage and sequestration by urban trees in the USA". Environmental Pollution 116 (2002) 381–389
Another area of the urban forest carbon cycle that this paper does not analyze is urban soils. Sixty-one percent of the total carbon in non-urban forest ecosystems
in the USA is stored in the soil environment (Birdsey and Heath, 1995). The amount of carbon from trees that is retained in urban soils, its residence time, and the amount of carbon currently stored in these soils remains to be investigated. It is likely, however, that urban soils contain less carbon per hectare than forest soils due to lower carbon inputs and increased soil decomposition rates due to warmer air and soil temperatures (e.g. Pouyat et al., 1997).