Actually, there are a number of questions with answers which label edges...
This might give you some ideas:
\documentclass[tikz]{standalone}
\usepackage{forest}
\begin{document}
\begin{forest}
for tree={
circle,
black,
draw,
fill=blue!40,
}
[{}
[{}, edge label={node [midway, above left] {B}}
[{}, edge label={node [midway, right] {A}}
[{}, edge label={node [midway, right] {U}}
[,phantom]
[{}, edge label={node [midway, left] {E}}
[{}, edge label={node [midway, left] {R}}, label=below:Bauer
]
[,phantom]
]
[{}, edge label={node [midway, right] {M}}, label=below:Baum
]
[,phantom]
]
[,phantom]
]
[,phantom]
]
[{}, edge label={node [midway, right] {F}}
[{}, edge label={node [midway, right] {E}}
[{}, edge label={node [midway, right] {L}}
[{}, edge label={node [midway, right] {D}}, label=below:Feld
]
[,phantom]
]
[,phantom]
]
[,phantom]
]
[{}, edge label={node [midway, left] {H}}
[{}, edge label={node [midway, left] {A}}
[{}, edge label={node [midway, left] {H}}
[{}, edge label={node [midway, left] {N}}, label=below:Hahn
]
[,phantom]
]
[{}, edge label={node [midway, right] {U}}
[{}, edge label={node [midway, left] {S}}, label=below:Haus
]
[,phantom]
]
[,phantom]
]
[{}, edge label={node [midway, left] {O}}
[{}, edge label={node [midway, right] {F}}, label=below:Hof
]
]
[{}, edge label={node [midway, right] {U}}
[,phantom]
[{}, edge label={node [midway, left] {H}}
[{}, edge label={node [midway, right] {N}}, label=below:Huhn
]
]
[{}, edge label={node [midway, right] {N}}
[,phantom]
[{}, edge label={node [midway, right] {D}}, label=below:Hund
]
]
]
]
[{}, edge label={node [midway, above right] {K}}
[,phantom]
[{}, edge label={node [midway, right] {A}}
[,phantom]
[{}, edge label={node [midway, right] {T}}
[,phantom]
[{}, edge label={node [midway, right] {Z}}
[,phantom]
[{}, edge label={node [midway, right] {E}}, label=below:Katze
]
]
]
]
]
]
\end{forest}
\end{document}

tikz-qtreeorforestfor this. See e.g. How to typeset a prefix tree (aka trie) ; How can I make Tikz tree node arcs not cross over each other? but if you look for questions involving either of those packages you will find plenty more. – Alan Munn May 31 '14 at 17:17