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Can any tree on $n$ nodes be decomposed into a set of $O(\log n)$ caterpillars? If not, what is the maximum number of caterpillars required? Are there efficient algorithms for finding the decomposition?

Any papers on this topic will be highly appreciated.

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It seems this is false in general. Take the tree with a central node $x$, connected to a bunch of nodes $y_1,\dots,y_n$. Each node $y_i$ is additionally connected to a node $z_i$.

The total number of nodes is $N=2n+1$.

Any caterpillar in this tree can contain at most two $z$ nodes, and therefore, $n/2$ caterpillars are needed to cover it. So it is impossible to cover it with $O(log N)$ caterpillars.

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I would like to add to dkuper's answer that it is possible to split every tree into a hierarchy of caterpillars, whose depth is at most $O(\log n)$. This can be done with Tarjan's heavy-light edge decompostion. Just take the maximal heavy-edge paths as the spines of the caterpillars.

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