= Splitting tree
A tree is splitting if every node has two incompatible extensions. This property is additional to height and level-size bounds in a <kappa-tree>. A <cofinal branch> through a splitting tree of regular uncountable height produces an equally large antichain by taking off-branch extensions and proceeding beyond their divergence heights. Thus a splitting $\omega_1$-tree with countable antichains has no uncountable chains; the nonsplitting chain $\omega_1$ itself is a counterexample to omitting splitting.
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