Depth of a leaf in a prefix tree (Huffman Coding)
Consider a symbol $\sigma$ whose frequency is $f$. During the course of Huffman's algorithm, $\sigma$ is matched with other nodes, and the depth of $\sigma$ is the number of times this happens.

At the first time that $\sigma$ is matched, it is either the symbol with smallest frequency or with second smallest frequency. In the former case, it is merged into a new symbol whose frequency is at least $2f$. In the latter case, after $\sigma$ is merged with the symbol of smallest frequency, all symbols have frequency at least $f$, and so the next time that it is merged, the merged symbol will have frequency at least $2f$.

Summarizing, it takes at most two mergings to double the frequency of a symbol. If its original frequency is $f$, then this shows that the depth of $f$ is at most $2\lceil \log_2(1/f) \rceil = O(\log(1/f))$.