The independence complex of a tree is contractible if and only if it reduces to a path P_n (n ≡ 1 mod 3) by truncation moves at branching points.
Adamaszek,A note on independence complexes of chordal graphs and dismantling, The elec- tronic journal of combinatorics24(2)(2017), P2.34
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Contractible independence complexes of trees
The independence complex of a tree is contractible if and only if it reduces to a path P_n (n ≡ 1 mod 3) by truncation moves at branching points.