Robinson said the calls reporting burning trees have recently increased in frequency.
Now, tree service owner Greg West is offering a $1,000 reward for information that leads to charges.
"We just got hit by a hurricane, and the only thing that lived were the palm trees," West told The Galveston County Daily News. "I want to put the reward out, so it might change somebody's mind to turn them in."
A conviction for arson is a second-degree felony, which carries a possible jail term from two to 20 years and a $10,000 fine.
The trees that weren't consumed by flames will live, but it could take years for the burned trunks to heal, West said. Many trees on the island have already died in flooding from Hurricane Ike in September.
A 40-year-old man carrying a gas can with only a residual amount of gasoline was arrested early Tuesday after he ran from police, Robinson said.
"After we apprehended him, a report of a palm tree fire did occur," Robinson said. "We're thinking there are possibly several people doing this."
The man hasn't been charged with arson but was charged with fleeing police, a misdemeanor, Robinson said.
Most of the fires occur in the early morning hours, Robinson said.
One car parked close to a burning tree was damaged, and utility lines have been replaced.
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