Suppose that we have a set of $n$ labels $L = \{\ell_1, \ell_2, \ldots, \ell_n\}$ and a perfect rooted binary tree $T$ of height $r \leq n$. For my purpose, the tree containing a single node will be considered to have height $1$. Then, for every node $v$ in $T$, we assign it a label $\mathscr{L}(v)$ from $L$ such that every root-to-leaf path in $T$ contains $r$ distinct labels. For a root-to-leaf path $P$ in $T$, we define $\mathscr{L}(P)$ to be the set of labels obtained from the nodes in $P$.
I am interested in the following question: if we label $T$ in this manner, how large (as a function of $n$ and $r$) can we make the set $\{\mathscr{L}(P) : P \texttt{ is a root to leaf path in } T \}$? As an example, when $r = n$, then the cardinality of the set above can only be $1$ since every root-to-leaf path will contain all $n$ labels.