  In the 1970s M. Laczkovich posed the following problem: Let
$\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an
uncountable Polish space $X$ equipped with the pointwise ordering.
\[\text{Characterize the order types of the linearly ordered subsets of
$\mathcal{B}_1(X)$.} \]The main result of the present paper is a complete
solution to this problem.
  We prove that a linear order is isomorphic to a linearly ordered family of
Baire class $1$ functions iff it is isomorphic to a subset of the following
linear order that we call $([0,1]^{<\omega_1}_{\searrow 0},<_{altlex})$, where
$[0,1]^{<\omega_1}_{\searrow 0}$ is the set of strictly decreasing transfinite
sequences of reals in $[0, 1]$ with last element $0$, and $<_{altlex}$, the so
called \emph{alternating lexicographical ordering}, is defined as follows: if
$(x_\alpha)_{\alpha\leq \xi}, (x'_\alpha)_{\alpha\leq \xi'} \in
[0,1]^{<\omega_1}_{\searrow 0}$, and $\delta$ is the minimal ordinal where the
two sequences differ then we say that \[ (x_\alpha)_{\alpha\leq \xi} <_{altlex}
(x'_\alpha)_{\alpha\leq \xi'} \iff (\delta \text{ is even and }
x_{\delta}<x'_{\delta}) \text{ or } (\delta \text{ is odd and }
x_{\delta}>x'_{\delta}). \] Using this characterization we easily reprove all
the known results and answer all the known open questions of the topic.
