Prüfer sequence

In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n − 2, and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918. One can generate a labeled tree's Prüfer sequence by iteratively removing vertices from the tree until only two vertices remain. Specifically, consider a labeled tree T with vertices {1, 2, ..., n}. At step i, remove the leaf with the smallest label and set the ith element of the Prüfer sequence to be the label of this leaf's neighbour. The Prüfer sequence of a labeled tree is unique and has length n − 2. Both coding and decoding can be reduced to integer radix sorting and parallelized. Consider the above algorithm run on the tree shown to the right. Initially, vertex 1 is the leaf with the smallest label, so it is removed first and 4 is put in the Prüfer sequence. Vertices 2 and 3 are removed next, so 4 is added twice more. Vertex 4 is now a leaf and has the smallest label, so it is removed and we append 5 to the sequence. We are left with only two vertices, so we stop. The tree's sequence is {4,4,4,5}. Let {a[1], a[2], ..., a[n]} be a Prüfer sequence: The tree will have n+2 nodes, numbered from 1 to n+2. For each node set its degree to the number of times it appears in the sequence plus 1. For instance, in pseudo-code: Convert-Prüfer-to-Tree(a) 1 n ← length[a] 2 T ← a graph with n + 2 isolated nodes, numbered 1 to n + 2 3 degree ← an array of integers 4 for each node i in T do 5 degree[i] ← 1 6 for each value i in a do 7 degree[i] ← degree[i] + 1 Next, for each number in the sequence a[i], find the first (lowest-numbered) node, j, with degree equal to 1, add the edge (j, a[i]) to the tree, and decrement the degrees of j and a[i]. In pseudo-code: 8 for each value i in a do 9 for each node j in T do 10 if degree[j] = 1 then 11 Insert edge[i, j] into T 12 degree[i] ← degree[i] - 1 13 degree[j] ← degree[j] - 1 14 break At the end of this loop two nodes with degree 1 will remain (call them u, v).