You can try make sure thay you have minimal collisions in your hash table, by making it large enough, depending on number of children at each node. You will need to experiment a bit for this. Then you can just use a very simple hash function which is fast and simple for this context. This will make sure you reach your child node in 1 memory access and a simple computation. For example, in you case, if input are literally alphabet letters, you can simply have a 26 size pointer array where you simply map the character to its pointer. Depending on your input, it will be slightly different.
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