  We consider online algorithms for the $k$-server problem on trees of size
$n$. Chrobak and Larmore proposed a $k$-competitive algorithm for this problem
that has the optimal competitive ratio. However, the existing implementations
have $O\left(k^2 + k\cdot \log n\right)$ or $O\left(k(\log n)^2\right)$ time
complexity for processing a query, where $n$ is the number of nodes. We propose
a new time-efficient implementation of this algorithm that has $O(n)$ time
complexity for preprocessing and $O\left(k\log k\right)$ time for processing a
query. The new algorithm is faster than both existing algorithms and the time
complexity for query processing does not depend on the tree size.
