On trees of bounded degree with maximal number of greatest independent sets

Given $n$ and $d$, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of $n$-vertex trees of vertex degree at most $d$. We show that the extremal tree is unique for all even $n$ but uniqueness may fail for odd $n$; moreover, for $d=3$ and every odd $n\geq7$, there are exactly $\lceil(n-3)/4\rceil+1$ extremal trees. In the paper, the problem of searching for extremal $(n,d)$-trees is also considered for the $2$-caterpillars; i.e., the trees in which every vertex lies at distance at most $2$ from some simple path. Given $n$ and $d\in\{3,4\}$, we completely reveal all extremal $2$-caterpillars on $n$ vertices each of which has degree at most $d$. Illustr. 9, bibliogr. 10.