Let $\mathbb N_0^\ast$ be the set of all finite words/sequences over $\mathbb N_0$ and $\varepsilon$ the empty word. For a word $a=(a_1,\ldots,a_n)$ we define $\operatorname{len}a:=n$, $\Sigma a:=\sum_{k=1}^n a_k$, $\operatorname{dep}a:=\operatorname{len}a+\Sigma a$, $\max a := \max\{a_k:1\leq k\leq n\}$ and say $a\rightarrow b$ if $b=(a_1,\ldots,a_{n-1},a_n+1)$ or $b=(a_1,\ldots,a_n,0)$. Now $T:=(\mathbb N_0^\ast,\rightarrow)$ is the directed infinite binary tree rooted in $\varepsilon$ and $\operatorname{dep} a$ is the directed distance from $\varepsilon$ to $a$ in $T$. Assigning a score function $s:\mathbb N_0^\ast\rightarrow\mathbb R$ we can now ask to maximize $s(a)$ with $\operatorname{dep}a\leq N$ for some $N\in\mathbb N_0$.

For this particular question set $$s(a) := \sum_{k=1}^{\operatorname{len} a}a_k\cdot\left(1+\max a\right)^{\operatorname{len} a - k + 1}$$ (but feel free to post interesting scores in the comment section).

Now what is the the maximal value of $s(a)$ with $\operatorname{dep} a \leq N$ and how can it be archieved? Non-trivial lower bounds would be appreciated too if the question turns out to be too hard (in a computational sense).

My gut feeling is something like $(K,0,\ldots,0)$ (but what would $K$ be depending on $N$?) or some kind of diagonal solution maybe, but optimization is not my strong suit.

(An interesting variation for another question might be to make this a two-player game (either finite or infinite) and look for winning strategies, when each player can only maximize over the nodes they selected in their turns.)

If this is a known problem, or variation of one, I would appreciate pointers in the comments as well.