  A maximal repetition, or run, in a string, is a maximal periodic substring
whose smallest period is at most half the length of the substring. In this
paper, we consider runs that correspond to a path on a trie, or in other words,
on a rooted edge-labeled tree where the endpoints of the path must be a
descendant/ancestor of the other. For a trie with $n$ edges, we show that the
number of runs is less than $n$. We also show an asymptotic lower bound on the
maximum density of runs in tries:
$\lim_{n\rightarrow\infty}\rho_\mathcal{T}(n)/n \geq 0.993238$ where
$\rho_{\mathcal{T}}(n)$ is the maximum number of runs in a trie with $n$ edges.
Furthermore, we also show an $O(n\log \log n)$ time and $O(n)$ space algorithm
for finding all runs.
