As we know, a [*tree-decomposition*](https://en.wikipedia.org/wiki/Tree_decomposition) of a graph must have following features:


1. All vertices are covered
2. All edges are covered
3. The connectivity condition


I think using repeated nodes in *tree-decomposition* must be allowed by definition. Is it true?



Repeated nodes in tree-decomposition of a graph is allowed or not?
The basic definition of tree decompositioning allows it. If we are trying to minimize the graph diameter or node count, then obviously they are redundant, and for algorithmic calculations they can be eliminated.


From the Wikipedia definition in your link is the definition of **smooth**:



> 
> A tree decomposition $(X, T = (I, F))$ of treewidth $k$ is smooth, if:
> 
> 
> * $\forall i \in I : |X\_i| = k + 1$,
> * $\forall (i, j) \in F : |X\_i \cap X\_j| = k$
> 
> 
> 


This disallows repeated nodes, as $|X\_i \cap X\_j| = k+1$ if $X\_i=X\_j$.


