The fact that Nathan Radley fills the knothole in the tree with cement proves that the gum and pennies are from Boo Radley.
The first hint as to where the gum and pennies came from is the location of the tree. The tree is on the Radley lot.
As we came to the live oaks at the Radley Place I raised my finger to point for the hundredth time to the knot-hole where I had found the chewing gum, trying to make Jem believe I had found it there, and found myself pointing at another piece of tinfoil. (Ch. 4)
The tree is easily accessible to Boo Radley, and he knows that the children will find what is left in it because they have to pass by. Boo has been watching them, and the presents in the tree are his way of reaching out to them. He wants to make friends. The gifts left in the tree make it clear that he is watching the children. The soap dolls are a prime example.
You may be wondering if someone else in the Radley household could have left the objects in the tree. This is highly unlikely. We know this because of Nathan Radley’s reaction when he finds out that the children are getting the gifts. He cements the tree’s knothole, so nothing else can be left there.
“Mr. Radley, ah—did you put cement in that hole in that tree down yonder?”
“Yes,” he said. “I filled it up.”
“Why’d you do it, sir?”
“Tree’s dying. You plug ‘em with cement when they’re sick. You ought to know that, Jem.” (Ch. 7)
Nathan Radley found out that Boo was leaving things in the tree for Scout and Jem to find, and he closed the hole. This way, Boo could not leave anything else in the tree, and his connection and communication with the Finch children would be greatly inhibited. Since Boo was a recluse, he had no other way to reach out to them.
The incident of the tree is an interesting one. The children earlier proved interest in and empathy toward Boo Radley. He was lonely, and decided to make friends. His unusual way of doing it allowed the children to get to know him a little bit better. Nathan put a stop to it because most people would think a grown man leaving gifts for children was inappropriate.