Let $T$ be a labeled tree with the root $v$, such that:
$(i)$ The height of the tree is $x$,
$(ii)$ the degree of the vertex $v$ is $y-2$,
$(iii)$ the degree of each vertex, except the leaves and the vertex $v$ is $y$.
Let $T(x,y)$ be the the set of subtrees of $T$ such that:
$(i)$ each subtree in $T(x,y)$ has exactly $x+1$ vertices, and
$(ii)$ each subtree in $T(x,y)$ has the vertex $v$.
There are multiple $T(x,y)$ satisfying $(i)–(iii)$. I take the largest such $T(x,y)$. Can one find an approximation for $|T(x,y)|$?
Example. $|T(2,3)|=2$, $|T(3,3)|=5$.