We consider the problem of constructing a sparse suffix tree (or suffix array) for $b$ suffixes of a given text $T$ of size $n$, using only $O(b)$ words of space during construction time. Breaking the naive bound of $Ω(nb)$ time for this problem has occupied many algorithmic researchers since a different structure, the (evenly spaced) sparse suffix tree, was introduced by K{ä}rkk{ä}inen and Ukkonen in 1996. While in the evenly spaced sparse suffix tree the suffixes considered must be evenly spaced in $T$, here there is no constraint on the locations of the suffixes. We show that the sparse suffix tree can be constructed in $O(n\log^2b)$ time. To achieve this we develop a technique, which may be of independent interest, that allows to efficiently answer $b$ longest common prefix queries on suffixes of $T$, using only $O(b)$ space. We expect that this technique will prove useful in many other applications in which space usage is a concern. Furthermore, additional tradeoffs between the space usage and the construction time are given.