  Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs
of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order
$n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$
with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the
graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges
within its independent set, where `$\nabla$' means the join product. For
$n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and
$G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioab\u{a}, Desai and Tait
[SIAM J. Discrete Math. 37 (3) (2023) 2228--2239] showed that for $\ell\geq 6$
and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$
contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a
problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree
$F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with
$|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm
SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and
sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq
\rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with
these two equalities holding. Secondly, we characterize the spectral extremal
graphs for some specific trees and provide asymptotic spectral extremal values
of the remaining trees. In particular, we characterize the spectral extremal
graphs for all spiders, surprisingly, the extremal graphs are not always the
spanning subgraph of $G_{n,\ell}$.
