I've done some investigation into this. All money goes to Arbor Day Foundation, but they are not planting trees themselves and work with partners like the US Forest Service.
To estimate cost-effectiveness we need to know tree mortality rate and estimated lifespan. The first factor depends on the region and maintenance level. US Forest Service estimated a 3% mortality rate per year. It takes around 40 years for a tree to absorb a tonne of CO2, and at least 30% of planted trees will survive to this age (0.97^40 ~ 30%). It's harder to estimate lifespan because trees can live hundreds and thousands of years. But it's less relevant if we want to reduce CO2 in the atmosphere in this century, not in the distant future. From these calculations, I can conclude that the cost-effectiveness of this program in the next 40 years must be around 3.3$ per tonne CO2 absorbed or cheaper. But it's just my back-of-the-envelope calculation for those who want some numbers. For example, Founders Pledge recommended intervention by The Coalition for Rainforest Nations avert a tonne of CO2 for 0.12$ with a plausible range $0.02-$0.72 (report). And these emissions will be averted right now, not in the future when trees grow up.
Overall, I stayed neutral to this campaign. On the one hand, I feel that it bring a lot of counterfactual dollars that would not be spent on charity otherwise. On the other, I think that paying too much attention to tree planting can take us away from the importance of reducing emissions.