# Caterpillars and alternating paths

Let p(m) (respectively, q(m)) be the maximum number k such that any tree with m edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with k edges. We derive closed-form expressions for p(m) and q(m) for all m ≥ 1. The two functions p(n) and q(n) can also be interpreted in terms of alternating paths among n disjoint line segments in the plane, whose 2n endpoints are in convex position.

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