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Returns the Prüfer sequence of the given tree.

A Prüfer sequence is a list of n - 2 numbers between 0 and n - 1, inclusive. The tree corresponding to a given Prüfer sequence can be recovered by repeatedly joining a node in the sequence with a node with the smallest potential degree according to the sequence.


T (NetworkX graph) – An undirected graph object representing a tree.


The Prüfer sequence of the given tree.

Return type:


  • NetworkXPointlessConcept – If the number of nodes in T is less than two.
  • NotATree – If T is not a tree.
  • KeyError – If the set of nodes in T is not {0, …, n - 1}.


There is a bijection from labeled trees to Prüfer sequences. This function is the inverse of the from_prufer_sequence() function.

Sometimes Prüfer sequences use nodes labeled from 1 to n instead of from 0 to n - 1. This function requires nodes to be labeled in the latter form. You can use relabel_nodes() to relabel the nodes of your tree to the appropriate format.

This implementation is from [1] and has a running time of \(O(n \log n)\).


[1]Wang, Xiaodong, Lei Wang, and Yingjie Wu. “An optimal algorithm for Prufer codes.” Journal of Software Engineering and Applications 2.02 (2009): 111. <https://doi.org/10.4236/jsea.2009.22016>


There is a bijection between Prüfer sequences and labeled trees, so this function is the inverse of the from_prufer_sequence() function:

>>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
>>> tree = nx.Graph(edges)
>>> sequence = nx.to_prufer_sequence(tree)
>>> sequence
[3, 3, 3, 4]
>>> tree2 = nx.from_prufer_sequence(sequence)
>>> list(tree2.edges()) == edges