It's no coincidence that stacks have a close relationship with trees.
You can get rid of any stacks whatsoever by keeping a parent pointer in the tree node data structure.
I wouldn't say you "get rid of any stacks" here.
While you don't implement the stack as a list in classical ways, the list of "parent pointer(s)" between the root and the current node at any given moment of the traversal forms a stack.
While traversing, you need no memory other than the current and the previous node/state.
"previous node/state" obviously refers to the "top of stack", the single point where a stack is accessible.
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