Identify this children's book from a vague explanation.

In kindergarten, I remember a series of books read to the class. There was probably even an entire K5 lesson plan created around these stories, or rather, it was likely designed specifically for a K5 lesson plan.

The main character, who I think was named “Mr. Peabody”, had a walking stick with his own face on it. The stick may have been magical or something, and maybe it could talk. I don’t remember. What I do remember is the partial origin of this stick. Mr Peabody noticed his own face growing out of this tree. And the face grew into a branch from which Peabody used to make his walking stick. So he’s got this stick with his own face on it. The teacher even had a “Peabody Stick” in class that had this dude’s face on it. Fuck, did I have a twisted ass kindergarten teach, or what??

Can anyone identify this story?