I'm trying to show that for every frequency vector $(p_1, p_2, p_3, p_4)$ such that $\sum_{i=1}^4 p_i=1$, the average word length outputted by Huffman algorithm is bounded at 2: If $(w_1,w_2,w_3,w_4)$ is the outputted code, then $\sum_{i=1}^4 p_i |w_i| \le 2$.
I've tried looking at the tree that is generated by Huffman algorithm, but the thing is that several different tree structures match different 4-sized frequency vectors and I can't tell something general about all of them.
Also, is there a more general theorem for $k, n$ (here $k=4, n=2$)?