Question

What is the Pru ̈fer code for a simple path whose vertices are numbered in increasing

order (i.e., 1 ∼ 2 ∼ 3 ∼ · · · ∼ n)?

Answer 1

Given that, the vertices are numbered in increasing order from 1 to n. The sequence will be having n-2 values.

The tree with vertices from 1 to n (Ex: 1 to 5) is as shown below and the prufercode will be having (n-2)=(5-2)=3 values.

Initially, the prufer code will be empty.

PruferCode={}

Now, start with the lowest label say a. Consider 2 as the lowest label in the above tree. Find the vertices connecting it to the rest of the tree say b. In this case it is connected to 1. Now remove a from the tree and add b to prufercode.

PruferCode={1}

Now, tree becomes,

Continue the same above process, until we are left with two nodes.

Now, the lowest label is 3, and it is connected to 1. So remove 3 and add 1 to prufercode.

PruferCode={1,1}

Now, tree becomes,

Now, the lowest label is 1, and it is connected to 5. So remove 1 and add 5 to prufercode.

PruferCode={1,1,5}

Now, tree becomes,

Stop the process now, because we left with only 2 nodes.

Therefore, the prufercode will be PruferCode={1,1,5}.