On the zeroth-order general Randi\'c index, variable sum exdeg index and trees having vertices with prescribed degree

The zeroth-order general Randi'c index (usually denoted by $R_{\alpha }^0$) and variable sum exdeg index (denoted by $SE{I}_a$) of a graph $G$ are defined as $R_{\alpha }^0(G)={\sum }_{v\in V(G)}(d_v{)}^{\alpha }$ and $SEI_{a}(G)= \sum_{v\in V(G)}d_{v}a^{d_{v}}$where$d_{v}$isdegreeofthevertex$v\in V(G)$,$a$isapositiverealnumberdifferentfrom1and$\alpha$ is a real number other than $0$ and $1$. A segment of a tree is a path $P$, whose terminal vertices are branching or pendent, and all non-terminal vertices (if exist) of $P$ have degree 2. For $n\ge 6$, let ${\mathbb{P}\mathbb{T}}_{n,n_1}$, ${\mathbb{S}\mathbb{T}}_{n,k}$, ${\mathbb{B}\mathbb{T}}_{n,b}$ be the collections of all $n$-vertex trees having $n_1$ pendent vertices, $k$ segments, $b$ branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randi'c index and variable sum exdeg index are determined from the collections ${\mathbb{P}\mathbb{T}}_{n,n_1}$, ${\mathbb{S}\mathbb{T}}_{n,k}$, ${\mathbb{B}\mathbb{T}}_{n,b}$. The obtained extremal trees for the collection ${\mathbb{S}\mathbb{T}}_{n,k}$ are also extremal trees for the collection of all $n$-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree $T$ can be determined from the number of vertices of $T$ with degree 2 and vise versa).