  On the space of rooted trees with possibly repeated labels, where all
vertices are unordered, we define two metrics: the best-match metric and the
left-regular metric. On the space of rooted trees with possibly repeated
labels, where vertices can be ordered or unordered, we cannot define a proper
metric. However, the best-match metric can be slightly modified to become a
semimetric on this space. To compute the best-match distance between two trees,
the expected time complexity and worst-case complexity are both
$\mathcal{O}(n^2)$, where $n$ is the tree size. To compute the left-regular
distance between two trees, the expected time complexity is $\mathcal{O}(n)$,
and the worst-case time complexity is $\mathcal{O}(n\log n)$. Such metrics
(semimetric) can be used to compare developmental trees, which describe
embryogenesis in developmental biology.
