we can fit this inside a tree so we have be able to fit all the code words inside a tree okay so two quick proofs we don't need to know to be able to prove going forward but I wanted to get across it these are actually relatively simple to prove consequence from this is probably something you've heard many many times and that will now be able to prove very very easily then for any message distribution P of X there's some distribution over symbols then you have an Associated uniquely decodable code see then the average encoding length so