# exercise on Huffman encode

I have these 4 symbols with their probabilities:

 x   P(x)
--------
1   0.3
2   0.3
3   0.2
4   0.2


I built the Huffman tree in this way:

and I obtainded:

 x   P(x)   C(x)
----------------
1   0.3     0
2   0.3     10
3   0.2     110
4   0.2     111


it's correct? Because according to the solution the results should be:

 x   P(x)   C(x)
----------------
1   0.3     00
2   0.3     01
3   0.2     10
4   0.2     11


Why? Yet I followed the steps shown here.

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## 1 Answer

A quick way to check whether your answer has a chance of being correct is to compute the average code length. Your encoding gives the average length of $2.1$, which is greater than using a code of fixed length $2$, so it can't be correct.

If you follow the priority queue algorithm from the source you cite, then you would notice that after merging nodes 3 and 4 you get one supernode of priority 0.4. Now your queue would have three elements of priorities $0.3, 0.3,$ and $0.4$. Thus, you would next merge elements corresponding to priorities $0.3$ and $0.3$ (the algorithm works by merging two nodes with lowest priorities), which happen to be nodes 1 and 2.