  The suffix trees are fundamental data structures for various kinds of string
processing. The suffix tree of a text string $T$ of length $n$ has $O(n)$ nodes
and edges, and the string label of each edge is encoded by a pair of positions
in $T$. Thus, even after the tree is built, the input string $T$ needs to be
kept stored and random access to $T$ is still needed. The \emph{linear-size
suffix tries} (\emph{LSTs}), proposed by Crochemore et al. [Linear-size suffix
tries, TCS 638:171-178, 2016], are a "stand-alone" alternative to the suffix
trees. Namely, the LST of an input text string $T$ of length $n$ occupies
$O(n)$ total space, and supports pattern matching and other tasks with the same
efficiency as the suffix tree without the need to store the input text string
$T$. Crochemore et al. proposed an \emph{offline} algorithm which transforms
the suffix tree of $T$ into the LST of $T$ in $O(n \log \sigma)$ time and
$O(n)$ space, where $\sigma$ is the alphabet size. In this paper, we present
two types of \emph{online} algorithms which "directly" construct the LST, from
right to left, and from left to right, without constructing the suffix tree as
an intermediate structure. Both algorithms construct the LST incrementally when
a new symbol is read, and do not access the previously read symbols. Both of
the right-to-left construction algorithm and the left-to-right construction
algorithm work in $O(n \log \sigma)$ time and $O(n)$ space. The main feature of
our algorithms is that the input text string does not need to be stored.
