I'm building a function that builds a dictionary with words, such as:

{'b': ['b', 'bi', 'bir', 'birt', 'birth', 'birthd', 'birthda', 'birthday'],
'bi': ['bi', 'bir', 'birt', 'birth', 'birthd', 'birthda', 'birthday'],
'birt': ['birt', 'birth', 'birthd', 'birthda', 'birthday'], 
'birthda': ['birthda', 'birthday'], 
'birthday': ['birthday'], 
'birth': ['birth', 'birthd', 'birthda', 'birthday'],
'birthd': ['birthd', 'birthda', 'birthday'], 
'bir': ['bir', 'birt', 'birth', 'birthd', 'birthda', 'birthday']}

This is what it looks like:

def add_prefixs(word, prefix_dict):
    lst=[]
    for letter in word:
        n=word.index(letter)
        if n==0:
            lst.append(word[0])
        else:
            lst.append(word[0:n])
    lst.append(word)
    lst.remove(lst[0])
    for elem in lst:
        b=lst.index(elem)
        prefix_dict[elem]=lst[b:]
    return prefix_dict

It works great for words like "birthday", but when I have a letter that repeats itself, I have a problem... for example, "hello".

{'h': ['h', 'he', 'he', 'hell', 'hello'], 'hell': ['hell', 'hello'], 'hello': ['hello'], 'he': ['he', 'he', 'hell', 'hello']}

I know it's because of the index (python chooses the index of the first time the letter appears) but I do not know how to solve it. Yes, this is my homework and I'm really trying to learn from you guys :)