  A trie $\mathcal{T}$ is a rooted tree such that each edge is labeled by a
single character from the alphabet, and the labels of out-going edges from the
same node are mutually distinct. Given a trie $\mathcal{T}$ with $n$ edges, we
show how to compute all distinct palindromes and all maximal palindromes on
$\mathcal{T}$ in $O(n)$ time, in the case of integer alphabets of size
polynomial in $n$. This improves the state-of-the-art $O(n \log h)$-time
algorithms by Funakoshi et al. [PCS 2019], where $h$ is the height of
$\mathcal{T}$. Using our new algorithms, the eertree with suffix links for a
given trie $\mathcal{T}$ can readily be obtained in $O(n)$ time. Further, our
trie-based $O(n)$-space data structure allows us to report all distinct
palindromes and maximal palindromes in a query string represented in the trie
$\mathcal{T}$, in output optimal time. This is an improvement over an existing
(na\"ive) solution that precomputes and stores all distinct palindromes and
maximal palindromes for each and every string in the trie $\mathcal{T}$
separately, using a total $O(n^2)$ preprocessing time and space, and reports
them in output optimal time upon query.
