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It is a strange question on a book.

Give an example of a tree $T$ that does not satisfy the following property: If $v$ and $w$ are vertices in $T$, there is a unique path from $v$ to $w$.

I think it is rather strange because a tree $T$ is defined to be a simple graph and if $v$ and $w$ are vertices in $T$,there is a unique simple path from $v$ to $w$. Maybe the answer is the difference between "path" and "simple path"? Thanks for your help.

ps: It is the 39th exercise in 7.1 Exercises of the book Discrete Mathematics (Fifth Edition) written by Richard Johnsonbaugh.

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Which book? $ $ –  Did Apr 7 '12 at 8:19
    
either their trees can have several connected components (what we might consider a "forest") or, more likely (as you say), the paths are not simple –  bgins Apr 7 '12 at 8:24
    
@Didier It is the 39th exercise in 7.1 Exercises of the book Discrete Mathematics (Fifth Edition) written by Richard Johnsonbaugh.Maybe it is a publishing mistake? –  tamlok Apr 7 '12 at 8:24
    
The author asks a tree to be connected hence the answer to your question is that a path can be not simple. For example the path 0-1-2-1-2-3 from 0 to 3 on the tree Z. –  Did Apr 7 '12 at 8:30
1  
If the path is allowed to be non-simple, then no tree has the property that there is a unique path joining each pair of vertices... –  Mariano Suárez-Alvarez Apr 7 '12 at 8:33
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