Why did Tom and Huck go to the old tree in The Adventures of Tom Sawyer?

Tom decides he wants to dig for a buried treasure left behind by pirates. He looks for a companion. When some of his first choices fail him, he approaches Huck. Huck, who is always up for an adventure, joins in.
Huck asks Tom how he knows where to dig, as the boys don't have a treasure map. Tom says:

"They always bury it under a ha’nted house or on an island, or under a dead tree that’s got one limb sticking out. Well, we’ve tried Jackson’s Island a little, and we can try it again some time; and there’s the old ha’nted house up the Still-House branch, and there’s lots of dead-limb trees—dead loads of ’em."

Hucks wonders, with all the dead trees to choose from, how they will decide which one to dig under for the treasure. Tom suggests the dead tree by the Still-House. The two boys dig there with no luck, and then Huck suggests the dead tree on Cardiff Hill:

"I reckon maybe we’ll tackle the old tree that’s over yonder on Cardiff Hill back of the widow’s."

Hucks worries, however, that since the old tree is on the Widow Douglas's property, she would claim any treasure they unearthed. Tom reassures him that any treasure they find will belong to them.
In short, Tom and Huck go to the old tree seeking a buried treasure left by pirates.