  This paper studies the stability and strategy-proofness aspect of the
two-sided one-to-one matching market. Agents have single-peaked preferences on
trees. In this setting, we characterize all rich anonymous tree-single-peaked
domains where a stable and (weakly group) strategy-proof matching rule exists.
We also show that whenever there exists a stable and strategy-proof matching
rule on a rich anonymous tree-single-peaked domain, one or both of the deferred
acceptance rules (Gale and Shapley, 1962) satisfy stability and weak group
strategy-proofness on that domain. Finally, we show that for markets with a
size of at least five, there is no rich anonymous domain where a stable and
non-bossy matching rule exists. As a corollary, we show incompatibility between
stability and group strategy-proofness on rich anonymous tree-single-peaked
domains for markets with a size of at least five.
