Two Metrics on Rooted Unordered Trees with Labels
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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 𝒪(n^2), where n is the tree size. To compute the left-regular distance between two trees, the expected time complexity is 𝒪(n), and the worst-case time complexity is 𝒪(nlog n). Such metrics (semimetric) can be used to compare developmental trees, which describe embryogenesis in developmental biology.
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