  We investigate the linear-chain structures in highly excited states of
$^{14}$C using a generalized molecular orbital model, by which we incorporate
an asymmetric configuration of three $\alpha$ clusters in the linear-chain
states. By applying this model to the $^{14}$C system, we study the
$^{10}$Be+$\alpha$ correlation in the linear-chain state of $^{14}$C. To
clarify the origin of the $^{10}$Be+$\alpha$ correlation in the $^{14}$C
linear-chain state, we analyze linear $3\alpha$ and $3\alpha+n$ systems in a
similar way. We find that a linear 3$\alpha$ system prefers the asymmetric
2$\alpha+\alpha$ configuration, whose origin is the many body correlation
incorporated by the parity projection. This configuration causes an asymmetric
mean field for two valence neutrons, which induces the concentration of valence
neutron wave functions around the correlating 2$\alpha$. A linear-chain
structure of $^{16}$C is also discussed.
