While reading about Esperanto grammar on this site, I came across the word "ligna" which appeared to mean "wooden". Why does Esperanto have "ligno" and "arbo" that have no common roots like "wood" and "tree" in English? Is it possible to derive a word from "arbo" to say "wooden"?

Quite likely the reason is that among those languages Zamenhof mastered, many (all?) have separate terms for tree (the plant) and wood (the substance). But yes, this is confusing at least to me, whose mother tongue does not usually make the difference. (Note, that English word "woods" has yet another meaning.)

Of course there is the adjective arba,
 

• Sub la arba foliaro…, Under the tree leaves…
  
• Sur senarba monto levu… On a tree-less mountain you should rise…

But in any case arb/ always refers to the plant, while lign/ to the substance.