  We determine, for all $k\geq 6$, the typical structure of graphs that do not
contain an induced $2k$-cycle. This verifies a conjecture of Balogh and
Butterfield. Surprisingly, the typical structure of such graphs is richer than
that encountered in related results. The approach we take also yields an
approximate result on the typical structure of graphs without an induced
$8$-cycle or without an induced $10$-cycle.
