Suppose that we have a decision tree of height $r + 1$ that describes how to increment an $n$-bit integer in the range $[0, 2^n -1]$. That is, the internal nodes are labelled with a bit position that you read, then depending on its value, you either go left or right and proceed to read the next bit indicated. The leaf nodes are labelled with a set of bits that are flipped to perform the increment operation. Note that the sequence of integers may not be the traditional Standard Binary Code, but could be any cyclic sequence of the $2^n$ integers. A Gray Code is another example of a sequence that may be produced (which corresponds to a Hamiltonian cycle in the hypercube graph).
Now, consider the root to "right before" leaf paths as sets of size $r$ from a universe of size $n$. I am trying to argue the following claim: Suppose that the root node of our decision tree is labelled by index $i$. If we take the paths on the left side of the tree (when $i$ is read as $0$) such that bit $i$ is not flipped, together with the paths on the right side of the tree (when $i$ is read as $1$) such that bit $i$ is flipped, then I am interested in bounding the size of the common intersection of those sets. In particular, we know that every set will contain index $i$ since it sits at our root node, but I would like to show that these sets must also have another bit in common.
I know that I do not want to use the sunflower lemma, as these sets may not have the same pairwise intersection, but I believe that I may be interested in something similar that characterizes the global intersection of all of these sets). Perhaps it may help to view the problem as a hypergraph, though I am not sure of results that may be of use. The other thing to note is that these are not arbitrary sets, as they are derived from a decision tree for increment integers, so there is a lot of structure involved.
Any help would be greatly appreciated, whether it's some guidance on how to prove this, or perhaps references to potentially useful theorems.
Edit: If it helps, we can restrict our attention to when $r = n - 1$, though I am ultimately interested in the general case as well.