Number of 2-3 trees of depth 4
Here is one interpretation of this question. We want to count rooted trees with ordered children in which each internal node has either 2 or 3 children.

Denote by $T_d$ the number of 2-3 trees of depth at most $d$, so $T_0 = 1$ (the tree consisting of a single node) and $T_1 = 3$ (the tree consisting of a a single node; a root with two children; a root with three children). More generally, a 2-3 tree of depth at most $d$ is either just a single node or consists of a root with two or three children, each of which is a 2-3 tree of depth at most $d-1$. Therefore $T_d = 1 + T_{d-1}^2 + T_{d-1}^3$.

Finally, the number of 2-3 trees of depth exactly $d$ is $T_d - T_{d-1}$. For example, the number of 2-3 trees of depth exactly 1 is $T_1 - T_0 = 3-1 = 2$.