Full Idea
An open branch in a completed tableau will always yield an interpretation that verifies every formula on the branch.
Gist of Idea
A completed open branch gives an interpretation which verifies those formulae
Source
David Bostock (Intermediate Logic [1997], 4.7)
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.177
A Reaction
In other words the open branch shows a model which seems to work (on the available information). Similarly a closed branch gives a model which won't work - a counterexample.
Related Idea
Idea 7790 If an argument is invalid, a truth tree will indicate a counter-example [Girle]