Yes, C should be it...
On Earth, among the surest indications of sunspot cycles are believed to be the rate that trees grow, as seen in the rings visible in the cross sections of their trunks.
A. On Earth, among the surest indications of sunspot cycles are believed to be the rate that trees grow
- verb problem. Need IS instead of ARE and need at which to define rate...
B. On Earth, among the surest indications of sunspot cycles are, it is believed, the rate of tree growth
- again verb problem...
C. On Earth, the rate at which trees grow is believed to be among the surest indications of sunspot cycles
- good, keep this.
D. Among the surest indications on Earth of sunspot cycles, believed to be the tree growth rate
- sentence fragment... No main verb... all phrases...
E. Among the surest indications on Earth of sunspot cycles is believed to be the rate at which trees grow
- here last phrase should modify the cycles not the growth of trees...