Answer:
Replace the word with impatiently.
Step-by-step explanation:
The word, "impatiently", is a better fit considering the context. Earnestly means, "with sincere and intense conviction; seriously". Impatiently means "in a restlessly eager manner".
In the context, the character is not expressing conviction for anything. He seems to be in a panicked state because he was getting farther from the tree. Therefore, impatiently is a better fit.
Other options could be, hastily, or fretfully.