In this paper, we present a new data structure called the packed compact trie (packed c-trie) which stores a set $S$ of $k$ strings of total length $n$ in $n \logσ+ O(k \log n)$ bits of space and supports fast pattern matching queries and updates, where $σ$ is the size of an alphabet. Assume that $α= \log_σn$ letters are packed in a single machine word on the standard word RAM model, and let $f(k,n)$ denote the query and update times of the dynamic predecessor/successor data structure of our choice which stores $k$ integers from universe $[1,n]$ in $O(k \log n)$ bits of space. Then, given a string of length $m$, our packed c-tries support pattern matching queries and insert/delete operations in $O(\frac{m}α f(k,n))$ worst-case time and in $O(\frac{m}α + f(k,n))$ expected time. Our experiments show that our packed c-tries are faster than the standard compact tries (a.k.a. Patricia trees) on real data sets. As an application of our packed c-trie, we show that the sparse suffix tree for a string of length $n$ over prefix codes with $k$ sampled positions, such as evenly-spaced and word delimited sparse suffix trees, can be constructed online in $O((\frac{n}α + k) f(k,n))$ worst-case time and $O(\frac{n}α + k f(k,n))$ expected time with $n \log σ+ O(k \log n)$ bits of space. When $k = O(\frac{n}α)$, by using the state-of-the-art dynamic predecessor/successor data structures, we obtain sub-linear time construction algorithms using only $O(\frac{n}α)$ bits of space in both cases. We also discuss an application of our packed c-tries to online LZD factorization.