Imagine you have a trie where all words begin with the same prefix of length L. Now, reporting all words beginning with that prefix requires you to do a complete search on the trie, which will take time O(n), where n is the total number of nodes in the trie.
The challenge here is that searching for all words in a trie starting with some prefix might require you to search a number of nodes unrelated to the length of the prefix. It depends what nodes are in the trie.
If you know you're going to do searches like this often, you might want to consider using a Patricia trie (also called a radix trie). This is a trie where each edge can be labeled with multiple characters and nodes with one child and one parent aren't allowed. If you store the trie this way, then you can show that the time required to visit all the nodes corresponding to words in a particular subtrie is O(z), where z is the number of words found. The reason this works is that every non-leaf node in a Patricia trie has at least two children, so a subtrie containing z leaf nodes will have z internal nodes (it's a good exercise to check this), so you can discover all the leaves after just scanning O(z) nodes. The problem is that this will find the nodes for the words rather than listing off the words themselves. If you're okay with that, this can be a huge time saver.
