Bingo! I just found out.
I'll put everything in spoiler.
The rubber tree is on the small island in the top right corner of your map. There place where you found the treasure chest with the seahorse artifact.
Now all you need to do is sail there, talk to the two pirates there, and make them dig the tree up.
Then you attack Mcgillicutty again, but this time you talk to him and call his mother fat so he shoots your mast.
Then it's all a matter of talking to the mermaid at Jerkbait island and make her repair the mast with the rubber tree.
And then I will leave the rest up to you!